ch.11
Recall the formula for t-score
(ref)
Recall the formula to find s ^2
(ref)
df: =2000¸ α=.05 The critical value is:
+/- 1.96
Type the blank step 1. State the null hypothesis 2. Determine the alpha level 3. Determine the appropriate statistical test 4. Identify the critical values of the test statistic 5. _________ ____ ______ __________ 6. Decide to reject or fail H0 7. Interpret the decision in terms of the original claim.
Calculate the test statistic
Degrees of freedom (df) for the one-sample t-test is equal to
N - 1.
How is the t-distribution defined?
The distribution of all possible values of t for random sample means selected from the raw score population described by H0
When is a t-test used instead of a z-test?
When the population standard deviation is unknown
Less than, "<" is _____ tailed
left
What does the shape of any particular sampling distribution of a correlation coefficient depend upon?
df
Which of the following would increase the power of a significance test for correlation?
Changing the sample size from N = 25 to N = 100
Which of the following is not a step in hypothesis testing? Interpret the results Summarize the findings in words Conduct a literature review State the null hypothesis
Conduct a literature review
For a two-tailed test with a = 0.05, the tcrit value is
different for each df.
more than, ">" is ____ tailed
right
The __ statistic can be used to estimate σ_{x̅}
s
What is sx̅?
standard error of the mean
If we look up a z-score of 2.42 in the z-score table we see that the area between the mean and a z-score of 2.42 = .4922, which means that the area above z = .0078 (since the sum of both numbers must = .5, half of the distribution). Since this was a two-tailed test, we ________ those probabilities to get the P-value associated with the outcome. The two-tailed probability of the z-score = .0078 + .0078 = .0156. P = .02
combine
The degrees of freedom is literally:
df: n-1
For x̅ = 125 sx = 10 N = 26 U = 128 for a one-tailed test with Ha > 128 and a= 0.01
-1.530; 2.485
Using the data given below, perform a one-tailed t-test with Ha: u <74 . Use a = 0.05. (ref #20 on guide)
-3.21 < -1.771, p < 0.05
The Cultural Diversity Task Force sponsored six events last year (e.g., Awareness Luncheon) and six events with the same names this year. To see how well the attendance last year correlated with attendance this year, the chairperson ranked the number of people attending each event during each of the two years and calculated a correlation. The chairperson used the Spearman rank-order correlation coefficient (0.05) to determine if attendance this year at these events was greater than last year (a positive correlation). The correlation would have to be bigger than _______ to be "significant."
0.829
A two-tailed t-test was conducted for For x̅ = 97 sx = 10 N = 14 U = 100 Using a = 0.05, tcrit = 2.16 was compared to tobt. What is the correct 95% confidence interval for u?
A confidence interval should not be constructed because the t-test was not significant.
When establishing the proper confidence interval for interval estimation, which of the following should be employed?
A two-tailed value of t-crit
For x̅ = 23 sx = 4 N = 12 U = 20 What is the tobt?
3.35
EXAMPLE Q: In a recent study, Lepp, Barkley, and Karpinski (2014) reported that undergraduate students spend an average of 278.67 (standard error of the mean = 9.79) minutes per day using their cell phones. You wonder if cell phone usage at your university *differs* significantly from Lepp, Barkley, and Karpinski's (2014) findings. Suppose a recent survey of students at your university found that students spend 255 minutes per day using their cell phones. Let's use the steps outlined above to set up and test the appropriate null hypothesis. With the outcome being - 2.42, and the cut off point being -1.96, do we reject or retain?
Yes.
Which of the following assumptions is common to both the z-test and the one-sample t-test? a. Alpha is selected to be 0.01. b. The raw score population forms a normal distribution, and the population mean and standard deviation are known. c. The standard deviation of the raw score population is estimated by . d. There is one random sample of interval or ratio scores.
There is one random sample of interval or ratio scores.
Which of the following is one of the assumptions for hypothesis testing of the Pearson correlation coefficient?
There is random sampling of X-Y pairs.
What is the difference between Pearson's and Spearman's correlations when performing a hypothesis test?
How the respective sampling distributions for each correlation are conceptualized
how to increase the power of a study
if two tailed, divide by 2 under (sig) from the print out. if the alpha level is two tailed you multiply by 2, and find the value on the selected chart.
If the researcher is conducting a significance test to see if the result that is obtained is reliably less than the population parameter, the test would be ____-tailed because the outcome will be evaluated at the left tail of the sampling distribution
left
Requirements for: No Known The sample size is less than the n
t-test
Unless we use the correct tcrit from the t-distribution for the appropriate N,
the probability of making a Type I error will not equal a.
"Not equal to", (=/) is ___ ______
two
Which of the following is a correct statement of a null hypothesis?
x̄ = μ x-bar and mu both represent means and can be compared.
tobt for a single sample formula:
x̅ - u / sx̅
The null hypothesis in a two-tailed significance test of correlation states that
no correlation exists in the population.
Type the blank step 1. State the null hypothesis 2. Determine the alpha level 3. Determine the appropriate statistical test 4. Identify the critical values of the test statistic 5. Calculate the test statistic 6. Decide to reject or fail H0 7. Interpret the decision in terms of the original claim.
Determine the alpha level
Type the blank step 1. State the null hypothesis 2. Determine the alpha level 3. _________ ____ ______ _________ ______ 4. Identify the critical values of the test statistic 5. Calculate the test statistic 6. Decide to reject or fail H0 7. Interpret the decision in terms of the original claim.
Determine the appropriate statistical test
df: =10¸ α=.01 The critical value is:
+/- 3.17
If a sample mean has a value equal to μ, the corresponding value of t will be equal to
0.0
Two Tailed, alpha = 0.05, df = 19, what isthe critical?
2.09 (2.093)
A memory researcher is testing an experimental behavioral treatment to see it it helps or hurts the memory of Alzheimer's Syndrome patients. After 3 months of treatment, she tests the patient's memories using a standardized test, and the data is presented below. The population mean for this test is 50. Did the treatment affect the patients' memories? Use .01 as the level of significance. 43 56 78 76 58 54 44 38 44 71 83 90 Which statistical test should be used?
A t-test would be appropriate here as the population standard deviation is not known. The sample size is also less than 30, so this also makes a t-test appropriate t-test
Researchers report that on average freshman who live on campus gain 15 pounds during their first year in college. You survey 10 of your friends and find that the average weight gain the first year was 16.2 lbs (s.d. = 2.6). How does weight gain among your friends compare to the national average? Assume an alpha level of .05 What is the correct statistical decision?
Refer to chart. Fail to Reject H0.
For each reported single-sample z-statistic below, state whether the correct decision would be to reject H0 or fail to reject H0. One-tailed test with α = .05, P-value of outcome = .035
Reject
The population standard error of the mean (σ) is necessary for conducting .
The single-sample -test for means
What happens to the t-distribution as the sample size increases?
The t-distribution appears more and more like a normal distribution.
EXAMPLE Q: In a recent study, Lepp, Barkley, and Karpinski (2014) reported that undergraduate students spend an average of 278.67 (standard error of the mean = 9.79) minutes per day using their cell phones. You wonder if cell phone usage at your university *differs* significantly from Lepp, Barkley, and Karpinski's (2014) findings. Suppose a recent survey of students at your university found that students spend 255 minutes per day using their cell phones. Let's use the steps outlined above to set up and test the appropriate null hypothesis. Step Four: Identify the critical value of the test statistic
We are conducting a two-tailed test and have set our alpha level at .05. Let's look at the area under the standard normal curve: In the example above, the critical value of z is the z-score that cuts off 2.5% of the area under the standard normal, or z-score, distribution at each tail of the distribution. As we have seen previously, z-scores of -1.96 and +1.96 are the cut-off points for a two-tailed test with an alpha or probability level of .05.
The logic behind computing a confidence interval is to compute the highest and lowest values of a _____ mean that are not significantly different from the _____.
population; the current sample mean
df=20¸ α=.05 The critical value is:
+/- 2.09
What is Tcrit for a one-tailed test for a negative tobt when 30 participants are tested? Use a =0.05.
-1.699
Researchers report that on average freshman who live on campus gain 15 pounds during their first year in college. You survey 10 of your friends and find that the average weight gain the first year was 16.2 lbs (s.d. = 2.6). How does weight gain among your friends compare to the national average? Compute the test statistic.
1.38 (t-test)
What z-score corresponds to an two-tailed alpha level of .05?
1.96 Dividing the 5% into two tails results in 2.5% in each tail with cutoff scores of plus or minus 1.96.
Using the data given below, perform a two-tailed t-test to compare the sample mean to u= 35.5. use a= 0.05 (ref. #18 on guide)
2.601 > 2.160, Reject H0
Two Tailed, alpha = 0.05, df= 12, what is the critical? (closest to?)
2.61 (2.282)
For x̅ = 52 sx = 3 N = 20 U = 50 for a two-tailed test with a = 0.05
2.981; 2.093
A memory researcher is testing an experimental behavioral treatment to see it it helps or hurts the memory of Alzheimer's Syndrome patients. After 3 months of treatment, she tests the patient's memories using a standardized test, and the data is presented below. The population mean for this test is 50. Did the treatment affect the patients' memories? Use .01 as the level of significance. 43 56 78 76 58 54 44 38 44 71 83 90 Compute the test statistic. Report your answer to two decimal places.
2.2 The sample mean is 61.25, and sample standard deviation is 17.72. The standard error of the mean is 5.12 (the standard deviation divided by the square root of 12). The t statistic is found by dividing the difference between the sample and population means (61.25-50=11.25) by the standard error (5.12), 11.25/5.12 = 2.20.
If a researcher reports a one-sample t-test with df = 24, how many individuals participated in this study?
25
A two-tailed t-test was conducted for For x̅ = 97 sx = 5 N = 21 U = 100 Using a = 0.05, trite = 2.086 was compared to tobt. What is the correct 95% confidence interval for ?
86.57 <_ u <107.43
If your research hypothesis is that the higher a person's IQ, the longer they will live, then your null hypothesis will be that
IQ and longevity are not related The point of the null hypothesis is that it argues that two variables are unrelated, whereas the research hypothesis argues that there is a relationship (whose direction may or may not be specified). See "Stating the Hypotheses". NO DIFFERENCE. NO RELATION.
A researcher is using a special phonics reading technique to help 10-year-olds to read. She wants to know if the technique has helped. To assess reading comprehension, she selects a nationally known test of comprehension on which the national mean score for 10-year-olds is 50 with a standard deviation of 7.5. The mean of the sample of 50 students is 54. Does the special phonics reading technique increase comprehension? Use .05 as the level of significance. What is the correct statistical decision?
Reject H0 3.77 is over the critical value of 1.64.
A memory researcher is testing an experimental behavioral treatment to see it it helps or hurts the memory of Alzheimer's Syndrome patients. After 3 months of treatment, she tests the patient's memories using a standardized test, and the data is presented below. The population mean for this test is 50. Did the treatment affect the patients' memories? Use .01 as the level of significance.. 43 56 78 76 58 54 44 38 44 71 83 90 What is the correct statistical decision?
The critical values for t for alpha = .01, two tailed, are-3.106 and 3.106. As the test statistic is not more extreme than either critical value, it is NOT in the rejection region and the null hypothesis can not be rejected. Fail to Reject
EXAMPLE 2 A group of local fishermen is concerned that the small size of the trout they've been catching may indicate a problem with the health and welfare of trout in nearby Lake Mechamu. They decide to start recording the length of trout they catch and test whether or not, based on their sample, the trout in the lake *differ* significantly in size from previous years. The Department of Natural Resources reported five years ago that the average length of trout in the lake was 15.6 inches; the report did not include the standard deviation of the measurement. Members of the group fish for the next several weeks, procure a sample of 15 trout, and record the length of each: 10, 8, 6, 10, 15, 12, 14, 15, 18, 12, 17, 16, 9, 17, 18 Sam, a member of the fishermen's group who is taking Introduction to Statistics at his local community college, volunteers to carry out the data analysis, and works through the steps below. Step 1: State the claim and identify the null (H0) and alternative (H1) hypotheses.
The null hypothesis states that the average size of the fish that Sam's friends caught is equal to the population parameter of 15.6 inches. The alternative hypothesis states that the average size of the fish that they caught is not equal to 15.6 inches. H0 = 15.6 H1 ≠ 15.6
A statistical decision to fail to reject H, can be interpreted as _______.
The probability of the computed statistic is >.05
A researcher is using a special phonics reading technique to help 10-year-olds to read. She wants to know if the technique has helped. To assess reading comprehension, she selects a nationally known test of comprehension on which the national mean score for 10-year-olds is 50 with a standard deviation of 7.5. The mean of the sample of 50 dyslexic students is 54. Does the special phonics reading technique increase comprehension? Use .05 as the level of significance. What is the correct interpretation of the finding?
There is evidence that the phonics technique significantly increases comprehension.
A memory researcher is testing an experimental behavioral treatment to see it it helps or hurts the memory of Alzheimer's Syndrome patients. After 3 months of treatment, she tests the patient's memories using a standardized test, and the data is presented below. The population mean for this test is 50. Did the treatment affect the patients' memories? Use .01 as the level of significance.. 43 56 78 76 58 54 44 38 44 71 83 90 What is the correct interpretation of the finding?
There is no evidence that the treatment affected the patients' memories. As the decision was to fail to reject the null hypothesis, there is no evidence of a significant difference on the patients' memories as a result of the treatment.
The alternative hypothesis in a two-tailed significance test of correlation states that
a correlation exists in the population.
Which of the following statements about the correlation coefficient is true? a. One should not accept that a correlation coefficient represents a relationship unless it is significant. b. Unless a correlation coefficient is zero, it represents a relationship. c. Positive correlation coefficients tend to be significant more often than negative ones. d. Sampling error does not apply to the correlation coefficient.
a. One should not accept that a correlation coefficient represents a relationship unless it is
Which of the following is one of the assumptions of a one-sample t-test? a. The obtained scores are on an ordinal or interval scale. b. The population standard deviation is estimated by computing sx . c. The population standard deviation is known. d. The population distribution is skewed.
b. The population standard deviation is estimated by computing sx .
When we hear that a recent survey found 36% +/- 4% of adults had used an on-line dating service, the+/-4% is the
margin of error.
When we construct a 95% confidence interval, we are 95% sure that the
population mean falls within the interval.
Suppose you conduct an experiment with 20 subjects and the tobt turns out to be 0.69, which is not statistically significant. Which of the following is the correct way to report your results?
t(19) = 0.69, p > 0.05
Suppose you conduct an experiment with 24 subjects and your turn out to be 2.92, which is statistically significant. Which of the following is the correct way to report your results?
t(23) = 2.92, p < 0.05
For the scenarios below, indicate whether you would use a z-test or a t-test, and whether the test would be two-tailed, left-tailed, or right-tailed. A researcher is interested in whether the reaction time (RT) to a signal change of a sample of 24 student drivers is slower (greater RT) than the RT published in a study of experienced drivers.
t-test, right-tailed
A memory researcher is testing an experimental behavioral treatment to see it it helps or hurts the memory of Alzheimer's Syndrome patients. After 3 months of treatment, she tests the patient's memories using a standardized test, and the data is presented below. The population mean for this test is 50. Did the treatment affect the patients' memories? Use .01 as the level of significance. 43 56 78 76 58 54 44 38 44 71 83 90 The appropriate alternative hypothesis will be ____ .
two-tailed As the researcher is looking to see if the treatment "affects" patients' memories, a two-tailed alternative should be used. The word "affect" is vague, and although it is hoped that the treatment could improve memory, as an experimental treatment, it could be harmful.
standard scores which indicate a raw score's distance from the mean in standard deviation units.
z-scores
Compute the degrees of freedom and look up the critical values of t for each problem below. Round the critical value to two decimal places. α= .05; 2-tailed, n=18
DF: 17 TCRIT: 2.11
Compute the degrees of freedom and look up the critical values of t for each problem below. Round the critical value to two decimal places. α=.01; 2-tailed; n=24
DF: 23 TCRIT: 2.81 (closest to)
Type the blank step 1. State the null hypothesis 2. Determine the alpha level 3. Determine the appropriate statistical test 4. Identify the critical values of the test statistic 5. Calculate the test statistic 6. ________ ___ _______ ___ _______ 7. Interpret the decision in terms of the original claim.
Decide to reject or fail H0
For each reported single-sample z-statistic below, state whether the correct decision would be to reject H0 or fail to reject H0. One-tailed test with α = .01, z = -0.67
Fail to reject
For each reported single-sample z-statistic below, state whether the correct decision would be to reject H0 or fail to reject H0. Two-tailed test with α = .01, P-value of outcome = .035
Fail to reject
EXAMPLE 3: For a class project in her statistics class, a student wants to investigate whether students at the college tend to favor the number seven when picking numbers from a group. In a sample of 371 students, 45 chose the number seven when picking a number between one and twenty "at random." Does this provide convincing statistical evidence of bias in favor of the number seven, in that the proportion of students picking seven is significantly higher than 1/20 = .05? Use a significance level of .01. Step 5: Calculate the test statistic
Find p̂, the proportion of selections that are equal to the number 7. p̂ = x / n p̂ = 45 / 371 p̂ = .12. Next, to calculate z, write down the formula and substitute the values of p̂, p, q, and n.
EXAMPLE 2 A group of local fishermen is concerned that the small size of the trout they've been catching may indicate a problem with the health and welfare of trout in nearby Lake Mechamu. They decide to start recording the length of trout they catch and test whether or not, based on their sample, the trout in the lake differ significantly in size from previous years. The Department of Natural Resources reported five years ago that the average length of trout in the lake was 15.6 inches; the report did not include the standard deviation of the measurement. Members of the group fish for the next several weeks, procure a sample of 15 trout, and record the length of each: 10, 8, 6, 10, 15, 12, 14, 15, 18, 12, 17, 16, 9, 17, 18 Sam, a member of the fishermen's group who is taking Introduction to Statistics at his local community college, volunteers to carry out the data analysis, and works through the steps below. Step 4: Identify the critical value of the test statistic to indicate under what condition the null hypothesis should be rejected or not rejected. THE INFO NEEDED TO CALCULATE THE T-CRIT: the α-level or probability level, which was set at Step 2 as α = .05, the degrees of freedom, computed here as df = n - 1 = 14, and whether the hypothesis is one-tailed or two-tailed; the test is two-tailed
Now, refer to chart. The critical value of t for a two-tailed test with α = .05 and df = 14 is ± 2.14. The logic of hypothesis testing for a single-sample t-test is identical to that for a single-sample z-test. If the single-sample t -statistic we obtain is greater than the critical value of 2.14 or less than -2.14, then our decision must be to reject H0. If our single-sample t falls between the critical values, i.e. -2.14 < t < 2.14, then we must fail to reject H0.
For each reported single-sample z-statistic below, state whether the correct decision would be to reject H0 or fail to reject H0. Two-tailed test with α = .05, z = 2.36
Reject
Type the blank step 1. _______ ___ ______ _______ 2. Determine the alpha level 3. Determine the appropriate statistical test 4. Identify the critical values of the test statistic 5. Calculate the test statistic 6. Decide to reject or fail H0 7. Interpret the decision in terms of the original claim.
State the null hypothesis
EXAMPLE Q: In a recent study, Lepp, Barkley, and Karpinski (2014) reported that undergraduate students spend an average of 278.67 (standard error of the mean = 9.79) minutes per day using their cell phones. You wonder if cell phone usage at your university *differs* significantly from Lepp, Barkley, and Karpinski's (2014) findings. Suppose a recent survey of students at your university found that students spend 255 minutes per day using their cell phones. Let's use the steps outlined above to set up and test the appropriate null hypothesis. Step Three: Determine the appropriate test to use.
The appropriate test for comparing a single-sample statistic to a population parameter when the mean and standard error of the mean are known is the single-sample z-test.
EXAMPLE 3: For a class project in her statistics class, a student wants to investigate whether students at the college tend to favor the number seven when picking numbers from a group. In a sample of 371 students, 45 chose the number seven when picking a number between one and twenty "at random." Does this provide convincing statistical evidence of bias in favor of the number seven, in that the proportion of students picking seven is significantly higher than 1/20 = .05? Use a significance level of .01. Step 4: Identify the critical value of the test statistic
The critical value for a right-tailed test is z = 2.33
EXAMPLE 3: For a class project in her statistics class, a student wants to investigate whether students at the college tend to favor the number seven when picking numbers from a group. In a sample of 371 students, 45 chose the number seven when picking a number between one and twenty "at random." Does this provide convincing statistical evidence of bias in favor of the number seven, in that the proportion of students picking seven is significantly higher than 1/20 = .05? Use a significance level of .01. Step 2: Specify the level of significance
We a re given 0.01, in this case, so a = 0.01
_______ _______ are those values of a standardized test statistic that cut off rejection (critical) regions of the distribution being used for the statistical test.
critical values
In the scenarios below, indicate whether single-sample hypothesis testing would be appropriate by responding "yes" or "no." Comparison of high school males and females on amount of marijuana smoked during summer vacation.
no
Researchers report that on average freshman who live on campus gain 15 pounds during their first year in college. You survey 10 of your friends and find that the average weight gain the first year was 16.2 lbs (s.d. = 2.6). How does weight gain among your friends compare to the national average? What is the correct statistical test?
single sample t-test
Entering 14 df and the t-score of -2.27, the calculator returns a two-tailed probability value of .0395. FYI, this result works out to be the same as if you were to take the one-tailed probability of .1977 and double it to get the probability for a two-tailed outcome. Are you getting the logic? According to APA style standards, the result of .0395 would be rounded to two significant digits and reported as "p = .__"
04
Sociologists want to test whether the number of homeless people in a particular urban area is increasing. In 2010, the average number of homeless people per day who sought shelter was 42.3 (σ = 6.2). Data from the current year reveal that the mean number of people seeking shelter per day is 45.3. Compute the test statistic using an alpha = .01 and n = 1000.
15.3 Since sigma is known use the z-statistic.
EXAMPLE 2 A group of local fishermen is concerned that the small size of the trout they've been catching may indicate a problem with the health and welfare of trout in nearby Lake Mechamu. They decide to start recording the length of trout they catch and test whether or not, based on their sample, the trout in the lake differ significantly in size from previous years. The Department of Natural Resources reported five years ago that the average length of trout in the lake was 15.6 inches; the report did not include the standard deviation of the measurement. Members of the group fish for the next several weeks, procure a sample of 15 trout, and record the length of each: 10, 8, 6, 10, 15, 12, 14, 15, 18, 12, 17, 16, 9, 17, 18 Sam, a member of the fishermen's group who is taking Introduction to Statistics at his local community college, volunteers to carry out the data analysis, and works through the steps below. Step 5: Calculate the test statistic.
1st, use the formula used to find the T-score. 2nd, Since Sam collected raw data, we have a little work to do to find the necessary elements for the single-sample t-test. We know that μ = 15.6. We also need to find the mean, x̅, and the standard deviation, s. To find s let's use the computational formula for the variance from Descriptive Statistics, shown here, then take the square root of the variance to find s, the standard deviation. Use the formula to find the s. Make an X2 chart Compute
EXAMPLE Q: In a recent study, Lepp, Barkley, and Karpinski (2014) reported that undergraduate students spend an average of 278.67 (standard error of the mean = 9.79) minutes per day using their cell phones. You wonder if cell phone usage at your university *differs* significantly from Lepp, Barkley, and Karpinski's (2014) findings. Suppose a recent survey of students at your university found that students spend 255 minutes per day using their cell phones. Let's use the steps outlined above to set up and test the appropriate null hypothesis. x̅, the sample mean, is?
225
EXAMPLE Q: In a recent study, Lepp, Barkley, and Karpinski (2014) reported that undergraduate students spend an average of 278.67 (standard error of the mean = 9.79) minutes per day using their cell phones. You wonder if cell phone usage at your university *differs* significantly from Lepp, Barkley, and Karpinski's (2014) findings. Suppose a recent survey of students at your university found that students spend 255 minutes per day using their cell phones. Let's use the steps outlined above to set up and test the appropriate null hypothesis. μ, the population mean, is?
278.67
A researcher is using a special phonics reading technique to help 10-year-olds to read. She wants to know if the technique has helped. To assess reading comprehension, she selects a nationally known test of comprehension on which the national mean score for 10-year-olds is 50 with a standard deviation of 7.5. The mean of the sample of 50 students is 54. Does the special phonics reading technique increase comprehension? Use .05 as the level of significance. Compute the test statistic. Report your answer to two decimal places.
3.77 First, find the standard error of the mean by dividing the population standard deviation by the square root of n, 7.5/ 7.07 =1.06. Then divide the difference between the means (54-50 = 4) by the standard error to get the z-value, 4/1.06 = 3.77. "See Large Sample Testing Using the z-test".
EXAMPLE Q: In a recent study, Lepp, Barkley, and Karpinski (2014) reported that undergraduate students spend an average of 278.67 (standard error of the mean = 9.79) minutes per day using their cell phones. You wonder if cell phone usage at your university *differs* significantly from Lepp, Barkley, and Karpinski's (2014) findings. Suppose a recent survey of students at your university found that students spend 255 minutes per day using their cell phones. Let's use the steps outlined above to set up and test the appropriate null hypothesis. σx̅ is the standard error of the mean, is?
9.79
EXAMPLE 4 According to an article published by the American Psychological Association in gradPSYCH Magazine, women receiving doctoral degrees in psychology outnumber men 3 to 1. In one social sciences statistics course, of the 25 psychology majors there are 18 women and 7 men. We want to know if the gender ratio of psychology majors in this course is different from the population gender ratio reported by the author. Both the sample data and the population data are proportions, not means. We can test whether a sample proportion differs from a population proportion using a single-sample z-test. The logic of the single-sample z-test for proportions is identical to the single-sample z-test for means, but the formula looks a bit different. The numerator of the z-statistic formula is the difference between the sample proportion and the population proportion. The denominator of this z-statistic estimates the standard error of the difference between the sample proportions under the assumption that the two population proportions are equal, i.e., under the assumption that H0 is true. Step 1: State the claim and identify the null and alternative hypotheses.
According to the APA, the population proportion of women to men earning psychology doctorates is 3 to 1. In other words, for every 4 people who receive a doctorate in psychology, 3 of the 4 are women. Written as a percentage, 75% are women; written as a proportion, .75 are women. So the population proportion, symbolized by p, is equal to .75. The null hypothesis states the proportion in this class is equal to the population proportion, while the alternative states the proportion is not equal to the population proportion. H0: p = .75 H1: p ≠ .75
For each of the following, state an appropriate null hypothesis. 62% of car buyers prefer silver cars to red cars.
Consider whether the comparisons are between means or proportions and single-sample cases or two-sample cases. H0: P0: 0.62
For each of the following, state an appropriate null hypothesis. A sample of girls is expected to score higher than the national average of 42.3 on a standardized reading test.
Consider whether the comparisons are between means or proportions and single-sample cases or two-sample cases. H0: x̄ = 42.3
For each of the following, state an appropriate null hypothesis. x̄=19.13; μ=23
Consider whether the comparisons are between means or proportions and single-sample cases or two-sample cases. H0: x̄ = μ
Compute the degrees of freedom and look up the critical values of t for each problem below. Round the critical value to two decimal places. α=.05; 1-tailed; n=28
DF: 27 TCRIT: 1.70
EXAMPLE 4 According to an article published by the American Psychological Association in gradPSYCH Magazine, women receiving doctoral degrees in psychology outnumber men 3 to 1. In one social sciences statistics course, of the 25 psychology majors there are 18 women and 7 men. We want to know if the gender ratio of psychology majors in this course is different from the population gender ratio reported by the author. Both the sample data and the population data are proportions, not means. We can test whether a sample proportion differs from a population proportion using a single-sample z-test. The logic of the single-sample z-test for proportions is identical to the single-sample z-test for means, but the formula looks a bit different. The numerator of the z-statistic formula is the difference between the sample proportion and the population proportion. The denominator of this z-statistic estimates the standard error of the difference between the sample proportions under the assumption that the two population proportions are equal, i.e., under the assumption that H0 is true. Step 5: Calculate the test statistic
Find p̂, the proportion of women in the class. p̂ = x / n p̂ = 18 / 25 p̂= 0=.72 Next, to calculate z, write down the formula and substitute the values of p̂, p, q and n.
Type the blank step 1. State the null hypothesis 2. Determine the alpha level 3. Determine the appropriate statistical test 4. __________ _____ ________ ___ ___ ______ __________ 5. Calculate the test statistic 6. Decide to reject or fail H0 7. Interpret the decision in terms of the original claim.
Identify the critical values of the test statistic
Type the blank step 1. State the null hypothesis 2. Determine the alpha level 3. Determine the appropriate statistical test 4. Identify the critical values of the test statistic 5. Calculate the test statistic 6. Decide to reject or fail H0 7. ___________ ___ _______ ___ ______ ___ ___ _______ _____
Interpret the decision in terms of the original claim
z-scores can be positive or negative, but area and probability are always ...
positive
EXAMPLE 4 According to an article published by the American Psychological Association in gradPSYCH Magazine, women receiving doctoral degrees in psychology outnumber men 3 to 1. In one social sciences statistics course, of the 25 psychology majors there are 18 women and 7 men. We want to know if the gender ratio of psychology majors in this course is different from the population gender ratio reported by the author. Both the sample data and the population data are proportions, not means. We can test whether a sample proportion differs from a population proportion using a single-sample z-test. The logic of the single-sample z-test for proportions is identical to the single-sample z-test for means, but the formula looks a bit different. The numerator of the z-statistic formula is the difference between the sample proportion and the population proportion. The denominator of this z-statistic estimates the standard error of the difference between the sample proportions under the assumption that the two population proportions are equal, i.e., under the assumption that H0 is true. Step 7: Interpret the decision in terms of the original claim.
Interpret the decision in terms of the original claim. There is no evidence that the gender ratio in the class is different from the published population proportions (p̂ = .72, p = .75, z = -0.35, p = .3632).
Sociologists want to test whether the number of homeless people in a particular urban area is increasing. In 2010, the average number of homeless people per day who sought shelter was 42.3 (σ = 6.2). Data from the current year reveal that the mean number of people seeking shelter per day is 45.3. What is the correct statistical test?
It is a test to compare a single sample with a known set of parameters. Single-sample -test
In the scenarios below, indicate whether single-sample hypothesis testing would be appropriate by responding "yes" or "no." A researcher wants to determine whether Hispanic and Native American men have similar startle responses when presented with an unexpected stimulus event.
No
Sociologists want to test whether the number of homeless people in a particular urban area is increasing. In 2010, the average number of homeless people per day who sought shelter was 42.3 (σ = 6.2). Data from the current year reveal that the mean number of people seeking shelter per day is 45.3. What is the correct statistical decision?
Reject null hypothesis The test statistic falls in the rejection region, so reject Ho.
EXAMPLE 2 A group of local fishermen is concerned that the small size of the trout they've been catching may indicate a problem with the health and welfare of trout in nearby Lake Mechamu. They decide to start recording the length of trout they catch and test whether or not, based on their sample, the trout in the lake differ significantly in size from previous years. The Department of Natural Resources reported five years ago that the average length of trout in the lake was 15.6 inches; the report did not include the standard deviation of the measurement. Members of the group fish for the next several weeks, procure a sample of 15 trout, and record the length of each: 10, 8, 6, 10, 15, 12, 14, 15, 18, 12, 17, 16, 9, 17, 18 Sam, a member of the fishermen's group who is taking Introduction to Statistics at his local community college, volunteers to carry out the data analysis, and works through the steps below. Step 3: Determine the appropriate test to use.
Since Sam knows the population mean (μ), but does not know the population standard deviation (σ), and since the sample size is small (n =15), he determines that the appropriate test is the single-sample t-test.
EXAMPLE 3: For a class project in her statistics class, a student wants to investigate whether students at the college tend to favor the number seven when picking numbers from a group. In a sample of 371 students, 45 chose the number seven when picking a number between one and twenty "at random." Does this provide convincing statistical evidence of bias in favor of the number seven, in that the proportion of students picking seven is significantly higher than 1/20 = .05? Use a significance level of .01. Step 6: Compare the calculated value to the critical value and decide whether to reject the null hypothesis.
The calculated value of z exceeds the critical value of 2.33, so reject the null hypothesis.
In this scenario, the probability of obtaining a sample with a mean length of 13.3 inches, corresponding to a t-score of -2.27, is .04. Since this P-value is below the limit of .05 set by alpha, the null hypothesis can be.....
rejected
EXAMPLE 2 A group of local fishermen is concerned that the small size of the trout they've been catching may indicate a problem with the health and welfare of trout in nearby Lake Mechamu. They decide to start recording the length of trout they catch and test whether or not, based on their sample, the trout in the lake differ significantly in size from previous years. The Department of Natural Resources reported five years ago that the average length of trout in the lake was 15.6 inches; the report did not include the standard deviation of the measurement. Members of the group fish for the next several weeks, procure a sample of 15 trout, and record the length of each: 10, 8, 6, 10, 15, 12, 14, 15, 18, 12, 17, 16, 9, 17, 18 Sam, a member of the fishermen's group who is taking Introduction to Statistics at his local community college, volunteers to carry out the data analysis, and works through the steps below. Step 6: Compare the calculated statistic to the critical value and decide to reject or fail to reject the null hypothesis.
Since the obtained t-value of -2.27 exceeds the critical value of ±2.14, the statistical decision is to reject H0, p< .05.
EXAMPLE 2 A group of local fishermen is concerned that the small size of the trout they've been catching may indicate a problem with the health and welfare of trout in nearby Lake Mechamu. They decide to start recording the length of trout they catch and test whether or not, based on their sample, the trout in the lake differ significantly in size from previous years. The Department of Natural Resources reported five years ago that the average length of trout in the lake was 15.6 inches; the report did not include the standard deviation of the measurement. Members of the group fish for the next several weeks, procure a sample of 15 trout, and record the length of each: 10, 8, 6, 10, 15, 12, 14, 15, 18, 12, 17, 16, 9, 17, 18 Sam, a member of the fishermen's group who is taking Introduction to Statistics at his local community college, volunteers to carry out the data analysis, and works through the steps below. Step 7: Interpret the decision in terms of the original claim.
Since the single-sample t-test we computed to compare the average length of fish caught by the fishers to the population mean yielded a t-score of -2.27 we reject H0. There is support for the alternative hypothesis that the lengths are different. The average length of the fish that the fishermen caught in Lake Mechamu is significantly less than the population mean of 15.56 inches (M = 13.3, SD = 3.87), t(14) = -2.27, p = .04.
Using the P-value approach with a t-test brings up a difference between the ______ _________ ________ ______ and the t-table. ___ ________ _________ _______ (same term) gives probabilities associated with z-scores. The t-table does not give probabilities but instead gives critical values of t. Since the critical values used with t are different for each combination of df and alpha, it would require a huge number of tables to include all the possibilities. Thus the tables are condensed to include only the critical values.
The Standard Normal Table
EXAMPLE Q: In a recent study, Lepp, Barkley, and Karpinski (2014) reported that undergraduate students spend an average of 278.67 (standard error of the mean = 9.79) minutes per day using their cell phones. You wonder if cell phone usage at your university *differs* significantly from Lepp, Barkley, and Karpinski's (2014) findings. Suppose a recent survey of students at your university found that students spend 255 minutes per day using their cell phones. Let's use the steps outlined above to set up and test the appropriate null hypothesis. Step Six: Compare the calculated statistic to the critical value and decide whether to reject or fail to reject the null hypothesis.
The average number of minutes spent using a cell phone in the population was 278.67. The average daily cell phone usage in our hypothetical university sample was 255 minutes. Since the mean number of minutes in our sample is less than the population mean, we would expect a negative z-score, because subtracting a larger number from a smaller number results in a negative answer. In other words, the z-score, or one-number-statistic, which we will use to evaluate our hypothesis, confirms that our sample mean was below the expected population mean. We now need to determine if the sample mean is so far below the population mean that we would expect to draw a sample with a mean that low from the population less than 5% of the time by chance. This last sentence is very important for understanding hypothesis testing, so read it again carefully to make sure you understand the logic behind it.
EXAMPLE 4 According to an article published by the American Psychological Association in gradPSYCH Magazine, women receiving doctoral degrees in psychology outnumber men 3 to 1. In one social sciences statistics course, of the 25 psychology majors there are 18 women and 7 men. We want to know if the gender ratio of psychology majors in this course is different from the population gender ratio reported by the author. Both the sample data and the population data are proportions, not means. We can test whether a sample proportion differs from a population proportion using a single-sample z-test. The logic of the single-sample z-test for proportions is identical to the single-sample z-test for means, but the formula looks a bit different. The numerator of the z-statistic formula is the difference between the sample proportion and the population proportion. The denominator of this z-statistic estimates the standard error of the difference between the sample proportions under the assumption that the two population proportions are equal, i.e., under the assumption that H0 is true. Step 6: Compare the calculated value to the critical value and decide whether to reject the null hypothesis.
The calculated value of z does not exceed the critical value of -1.96, so fail to reject the null hypothesis.
EXAMPLE 3: For a class project in her statistics class, a student wants to investigate whether students at the college tend to favor the number seven when picking numbers from a group. In a sample of 371 students, 45 chose the number seven when picking a number between one and twenty "at random." Does this provide convincing statistical evidence of bias in favor of the number seven, in that the proportion of students picking seven is significantly higher than 1/20 = .05? Use a significance level of .01. Step 7: Interpret the decision in terms of the original claim.
The calculated value of z exceeds the critical value of 2.33, so reject the null hypothesis. There is evidence of bias in favor of the number seven. Specifically, the number 7 is chosen more often than expected by chance (p̂ = .12, p = .05, z = 6.19, p = .0001)
EXAMPLE 3: For a class project in her statistics class, a student wants to investigate whether students at the college tend to favor the number seven when picking numbers from a group. In a sample of 371 students, 45 chose the number seven when picking a number between one and twenty "at random." Does this provide convincing statistical evidence of bias in favor of the number seven, in that the proportion of students picking seven is significantly higher than 1/20 = .05? Use a significance level of .01. Step 1: State the claim and identify the null and alternative hypotheses.
The claim is that there is bias in favor of selecting the number seven. H0: p ≤ .05 H1: p > .05
EXAMPLE 4 According to an article published by the American Psychological Association in gradPSYCH Magazine, women receiving doctoral degrees in psychology outnumber men 3 to 1. In one social sciences statistics course, of the 25 psychology majors there are 18 women and 7 men. We want to know if the gender ratio of psychology majors in this course is different from the population gender ratio reported by the author. Both the sample data and the population data are proportions, not means. We can test whether a sample proportion differs from a population proportion using a single-sample z-test. The logic of the single-sample z-test for proportions is identical to the single-sample z-test for means, but the formula looks a bit different. The numerator of the z-statistic formula is the difference between the sample proportion and the population proportion. The denominator of this z-statistic estimates the standard error of the difference between the sample proportions under the assumption that the two population proportions are equal, i.e., under the assumption that H0 is true. Step 4: Identify the critical value of the test statistic.
The critical value for a two-tailed test is z = ±1.96
EXAMPLE Q: In a recent study, Lepp, Barkley, and Karpinski (2014) reported that undergraduate students spend an average of 278.67 (standard error of the mean = 9.79) minutes per day using their cell phones. You wonder if cell phone usage di *differs* significantly from Lepp, Barkley, and Karpinski's (2014) findings. Suppose a recent survey of students at your university found that students spend 255 minutes per day using their cell phones. Let's use the steps outlined above to set up and test the appropriate null hypothesis. Step one: State the claim include the symbolic form
There is a no difference between the reported study and the cell phone usage of undergraduate students. There is a difference between the reported study and the cell phone usage of undergraduate students H null = 278.67 H alternative ≠ 278.67 (which is the claim if you read the scenairo)
EXAMPLE Q: In a recent study, Lepp, Barkley, and Karpinski (2014) reported that undergraduate students spend an average of 278.67 (standard error of the mean = 9.79) minutes per day using their cell phones. You wonder if cell phone usage at your university *differs* significantly from Lepp, Barkley, and Karpinski's (2014) findings. Suppose a recent survey of students at your university found that students spend 255 minutes per day using their cell phones. Let's use the steps outlined above to set up and test the appropriate null hypothesis. Step Five: Calculate the test statistic.
Use the standard z-score formula for hypothesis testing of single samples. You should receive 2.42.
EXAMPLE Q: In a recent study, Lepp, Barkley, and Karpinski (2014) reported that undergraduate students spend an average of 278.67 (standard error of the mean = 9.79) minutes per day using their cell phones. You wonder if cell phone usage at your university *differs* significantly from Lepp, Barkley, and Karpinski's (2014) findings. Suppose a recent survey of students at your university found that students spend 255 minutes per day using their cell phones. Let's use the steps outlined above to set up and test the appropriate null hypothesis. *crucial: step seven: Interpret the decision in terms of the original claim*
We rejected H0 so the alternative hypothesis is supported, providing support to the claim that a difference in cell phone usage exists between the students at your university and the students in Lepp, Barkley, and Karpinski's (2014) study. The students at your university spend significantly fewer minutes per day on their cell phones than did students in Lepp, Barkley, and Karpinski's (2014) reference population.
EXAMPLE Q: In a recent study, Lepp, Barkley, and Karpinski (2014) reported that undergraduate students spend an average of 278.67 (standard error of the mean = 9.79) minutes per day using their cell phones. You wonder if cell phone usage at your university *differs* significantly from Lepp, Barkley, and Karpinski's (2014) findings. Suppose a recent survey of students at your university found that students spend 255 minutes per day using their cell phones. Let's use the steps outlined above to set up and test the appropriate null hypothesis. Step Two: Specify the level of significance.
We will set the signifance level, a = 0.05
In the scenarios below, indicate whether single-sample hypothesis testing would be appropriate by responding "yes" or "no." A sociologist wants to determine how the survival rate for newborns in a small developing country compares to the rate reported by World Health Organization for heavily industrialized countries.
Yes
In the scenarios below, indicate whether single-sample hypothesis testing would be appropriate by responding "yes" or "no." You wonder if the proportion of female to male psychology majors at your university is similar to that found in a national survey of university students.
Yes
Researchers report that on average freshman who live on campus gain 15 pounds during their first year in college. You survey 10 of your friends and find that the average weight gain the first year was 16.2 lbs (s.d. = 2.6). How does weight gain among your friends compare to the national average? What is the correct interpretation of the finding?
Your friends' freshman-year weight gain does not differ significantly from the national average
EXAMPLE 4 According to an article published by the American Psychological Association in gradPSYCH Magazine, women receiving doctoral degrees in psychology outnumber men 3 to 1. In one social sciences statistics course, of the 25 psychology majors there are 18 women and 7 men. We want to know if the gender ratio of psychology majors in this course is different from the population gender ratio reported by the author. Both the sample data and the population data are proportions, not means. We can test whether a sample proportion differs from a population proportion using a single-sample z-test. The logic of the single-sample z-test for proportions is identical to the single-sample z-test for means, but the formula looks a bit different. The numerator of the z-statistic formula is the difference between the sample proportion and the population proportion. The denominator of this z-statistic estimates the standard error of the difference between the sample proportions under the assumption that the two population proportions are equal, i.e., under the assumption that H0 is true. Step 2: Specify the level of significance.
a = 0.05
EXAMPLE 3: For a class project in her statistics class, a student wants to investigate whether students at the college tend to favor the number seven when picking numbers from a group. In a sample of 371 students, 45 chose the number seven when picking a number between one and twenty "at random." Does this provide convincing statistical evidence of bias in favor of the number seven, in that the proportion of students picking seven is significantly higher than 1/20 = .05? Use a significance level of .01. Step 3: Determine the appropriate test to use.
p = .05, the probability of selecting number 7 q = 1 - p = .95, the probability of not selecting 7 Both np and nq ≥ 5, so the sampling distribution for p̂ is approximately normal. Use the z-test for proportions.
EXAMPLE 4 According to an article published by the American Psychological Association in gradPSYCH Magazine, women receiving doctoral degrees in psychology outnumber men 3 to 1. In one social sciences statistics course, of the 25 psychology majors there are 18 women and 7 men. We want to know if the gender ratio of psychology majors in this course is different from the population gender ratio reported by the author. Both the sample data and the population data are proportions, not means. We can test whether a sample proportion differs from a population proportion using a single-sample z-test. The logic of the single-sample z-test for proportions is identical to the single-sample z-test for means, but the formula looks a bit different. The numerator of the z-statistic formula is the difference between the sample proportion and the population proportion. The denominator of this z-statistic estimates the standard error of the difference between the sample proportions under the assumption that the two population proportions are equal, i.e., under the assumption that H0 is true. Step 3: Determine the appropriate test to use.
p = .75, the population proportion of women q = 1 - p = .25, the population proportion of men Both np and nq ≥ 5, so the sampling distribution for p̂ is approximately normal. Use the z-test for proportions.
To summarize this scenario, the probability of obtaining a mean cell phone score of 255 minutes, corresponding to a z-score of -2.42 (or a score 2.42 standard deviations below the population mean) is .0156. Since this P-value is below the limit of .05 set by alpha, the null hypothesis can be _______; there is less than a .05 probability the results were due to chance. According to APA style standards, the result would be rounded to two significant digits and reported as "P = .02."
rejected
if the researcher is conducting a significance test to see if the result that is obtained is reliably greater than the population parameter, the test would be _____-tailed because we would be evaluating the outcome at the right tail of the sampling distribution.
right
For the scenarios below, indicate whether you would use a z-test or a t-test, and whether the test would be two-tailed, left-tailed, or right-tailed. A social worker intends to survey a sample of 20 households in a neighborhood in a predominantly low-income area in order to see if the average household income is above or below the poverty level figures published by the federal government.
t-test, two-tailed
If the alternative hypothesis does not predict a specific direction for the difference, an outcome in either tail (___)-tailedof the distribution of possible scores could be regarded as a "not-equal-to" result leading to rejection of H0.
two
A researcher is using a special phonics reading technique to help 10-year-olds to read. She wants to know if the technique has helped. To assess reading comprehension, she selects a nationally known test of comprehension on which the national mean score for 10-year-olds is 50 with a standard deviation of 7.5. The mean of the sample of 50 students is 54. Does the special phonics reading technique increase comprehension? Which test statistic should be used?
z- test A z-test would be appropriate here as the population standard deviation is known. The sample size is also greater than 30, so this also makes a z-test appropriate.
Requirements for: A known that is evenly distributed, or The Sample size is greater or equal to the n
z-Test
For the scenarios below, indicate whether you would use a z-test or a t-test, and whether the test would be two-tailed, left-tailed, or right-tailed. A clinical psychologist wants to compare the scores of his therapy group of 18 clients on a measure of self-confidence to the results in a published paper that includes the mean and standard deviation.
z-test, two-tailed
Another term that means probability level is .
α
EXAMPLE 2 A group of local fishermen is concerned that the small size of the trout they've been catching may indicate a problem with the health and welfare of trout in nearby Lake Mechamu. They decide to start recording the length of trout they catch and test whether or not, based on their sample, the trout in the lake differ significantly in size from previous years. The Department of Natural Resources reported five years ago that the average length of trout in the lake was 15.6 inches; the report did not include the standard deviation of the measurement. Members of the group fish for the next several weeks, procure a sample of 15 trout, and record the length of each: 10, 8, 6, 10, 15, 12, 14, 15, 18, 12, 17, 16, 9, 17, 18 Sam, a member of the fishermen's group who is taking Introduction to Statistics at his local community college, volunteers to carry out the data analysis, and works through the steps below. Step 2: Determine alpha level
α = .05.
