PSY 292 Exam 2
Practice: For a normal population the sample mean M = 35 has the same z-score as the sample mean M = 43. What is the population mean?
39
Practice: If all the possible random samples of size n = 7 are selected from a population with μ = 70 and σ = 5 and the mean is computed for each sample, then what value will be obtained for the mean of all the sample means?
70
For a sample size of 9 with SS = 100 and sample mean M = 18.2, what is the probability that the true population mean will be between 16.56 and 19.84?
80%
If it asks for mean of all sample means, how would you find it?
= μ
What are the 4 assumptions of hypothesis tests with z scores?
(1) random sampling, (2) independent observations, (3) the standard deviation is unchanged by the treatment, (4) normal sampling distributions
Practice: What is the z-score for a sample mean of M = 21 where the population mean is 24, the population standard deviation is 3, and the sample size is 16?
-4
Which of the following is a common limitation of hypothesis testing?
-Conclusions are made about the data set rather than about the hypothesis itself. -Demonstrating a significant treatment effect does not necessarily indicate a substantial treatment effect.
For a repeated-measures t test, the cutoff value for α = _____ using a one-tailed test is the same as the cutoff value for α = _____ using a two-tailed test.
0.025, 0.05
Practice: A beetle population has a mean length of 2.3 inches with standard deviation of 0.2 inches. What is the standard error for a sample size of 16?
0.05
What affects significance?
1. Variability in scores 2. Sample size
For a repeated-measures study comparing two treatments with 12 scores in each treatment, what is the df value for the t statistic?
11
Practice: A researcher is working with a population of data. He needs the standard distance between sample mean and the population mean to be at most 1.3. If the standard deviation of the population is 5, how large will the sample size need to be?
15
Practice: A population has a mean of 120 and a standard deviation of 10. If a sample of size 25 is collected, how much distance on average is expected between the sample mean and the population mean?
2
Practice:The population of scores on a nationally standardized test forms a normal distribution with μ = 300 and σ = 50. If you take a random sample of n = 25 students, what is the probability that the sample mean will be less than M = 280?
2.28% 1st=find σx̅ 2nd= find z score Z=(M-μ)/σx̅ 3rd=match z score on statistic table under .00 for number/% 4th=subtract % from table from 100
A researcher is looking at the impact that television has on children. Children are placed in a room with a variety of toys and a television playing a cartoon. The researcher predicts that the children will spend more than half of their 30 minutes looking at the television. The researcher tested 15 children and found a sample mean of M = 17 minutes spent watching the television with SS = 79. In order to test this hypothesis, what does the researcher need?
A one-tailed t statistic
When is a one-tailed test used?
A one-tailed test is used when there is a specific interest in one direction on of influence: either increase or decrease.
how can you describe the standard error (σx̅)?
standard distance between the sample mean and the population mean
What is the formula for Cohen's d ?
Cohen's d= (M-μ)/σ
standard distance between the sample mean and the population mean
standard error (σx̅)
what happens when sample size increases?
standard error decreases
what happens when sample size decreases
standard error increases
What is the null hypothesis for a related-samples test?
H0: μD = 0
If the standard error among sample means is small, which of the following is true? I. All the possible sample means are clustered close together. II. A researcher can be confident that any individual sample mean will provide a reliable measure of the population. III. A researcher must be concerned that a different sample could produce a different conclusion.
I and II
Practice: Which of the following is a true statement for any population with mean μ and standard deviation σ? I. The distribution of sample means for sample size n will have a mean of μ. II. The distribution of sample means for sample size n will have a standard deviation of. III. The distribution of sample means will approach a normal distribution as n approaches infinity.
I, II, and III
Why might a repeated-measures study require half the number of subjects compared to a similar matched-subjects study with the same number of scores?
In the repeated-measures study, each subject could be measured twice
If a researcher is concerned that a standard error is too big for the sample mean to provide a reliable measure of the population mean, what can the researcher do?
Increase the sample size.
What happens to the standard distance between the sample mean and the population mean when the sample size is multiplied by 4?
It is divided by 2.
A researcher failed to reject the null hypothesis with a two-tailed test using α = .05. If the researcher had used the same data with a one-tailed test, what can we conclude?
It is impossible to tell whether or not the researcher would reject the null hypothesis using a one-tailed test
Practice: All of the possible random samples of size 5 are selected from a population and the variance among these sample means is 16. If all possible random samples of size 10 are selected from the same population, what can we say about the variance of this new set of sample means?
It will be less than 16
The standard error for a particular sample is 3.6. A researcher needs the standard error to be 1.2. What should the researcher do to the sample size?
Multiply the sample size by 9
For a repeated-measures study comparing two treatments with n = 26 scores in each treatment, the data produce t = 2.13. If the mean difference is in the direction that is predicted by the researcher, then which of the following is the correct decision for a hypothesis test with α = .05?
Reject H0 for either a one-tailed test or a two-tailed test
What is sampling error?
The natural error that exists between a sample and its corresponding population
In measuring the effect of hours of sleep on performance on a mathematics test using a repeated-measures study, which of the following is an example of an order effect? a. The participants' moods may be different during the first assessment than during the second, thereby affecting performance. b. The participants may have gained experience in taking the test the first time, thereby making it difficult to determine whether the change in hours of sleep or the experience causes the difference in performance. c. Because the participants were ordered to take the assessments, this may negatively impact their performance. d. The order in which participants finish the test may be the factor that impacts their performance rather than the hours of sleep.
The participants may have gained experience in taking the test the first time, thereby making it difficult to determine whether the change in hours of sleep or the experience causes the difference in performance.
A researcher administers a treatment to a sample of n = 100 participants and uses a hypothesis test to evaluate the effect of the treatment. The hypothesis test produces a z-score of z = 2.1. Assuming that the researcher is using a two-tailed test, what should the researcher do?
The researcher should reject the null hypothesis with α = .05, but not with α = .01
What is a Type II error?
failing to reject a false null hypothesis
What is not a characteristic of the distribution of sample means? a. The sample means should pile up around the population mean. b. The pile of sample means should tend to form a normal-shaped distribution. c. The sample means should have similar standard deviations as the population standard deviation. d. The larger the sample size, the closer the sample means should be to the population mean.
The sample means should have similar standard deviations as the population standard deviation.
Which quantity decreases as the sample size increases?
The standard error
The results of a hypothesis test with a repeated-measures t statistic are reported as follows: t(9) = 2.28, p < .05. Which of the following is consistent with the report?
The study used a total of 10 participants, and the mean difference was not significant.
A researcher is conducting a directional (one-tailed) test with a sample of n = 10 to evaluate the effect of a treatment that is predicted to increase scores. If the researcher obtains t = 2.770, then what decision should be made?
The treatment has a significant effect with α = .05 but not with α = .01.
What is the purpose of matching subjects on a variety of variables in a matched-subjects design?
To reduce or eliminate the effect of these variables on the variable that is being studied
s²=SS/n-1
Variance formula for population variance
What is a Type I error?
When a researcher rejects a null hypothesis that is actually true
When is the distribution of sample means identical to the population distribution?
When n = 1 When the standard error equals the population standard deviation
What is another name for a repeated-measures design?
Within-subjects design
Z score used for locating a sample mean
Z=(M-μ)/σx̅
A researcher is evaluating the influence of a treatment using a sample selected from a normally distributed population with a mean of μ = 30 and a standard deviation of σ = 3. The researcher expects a 1-point treatment effect and plans to use a two-tailed hypothesis test with α = 0.05. Compute the power of the test if the researcher uses n = 9 individuals. a. 17% b. 50% c. 83% d. Nearly 100%
a. 17%
Which of the following correctly describes the effect that decreasing sample size and decreasing the standard deviation have on the power of a hypothesis test? a. A decrease in sample size will decrease the power, but a decrease in standard deviation will increase the power. b. Both will increase the power. c. A decrease in sample size will increase the power, but a decrease in standard deviation will decrease the power d. Both will decrease the power.
a. A decrease in sample size will decrease the power, but a decrease in standard deviation will increase the power.
Which of the following is an advantage that an independent-measures study has over a repeated-measures study? a. An independent-measures design can eliminate time-related factors. b. An independent-measures design can use fewer participants. c. An independent-measures design can be used on populations with large variances. d. There is no advantage to using an independent-measures study. Hide Feedback
a. An independent-measures design can eliminate time-related factors.
A researcher expects a treatment to produce a decrease in the population mean. The treatment is evaluated using a one-tailed hypothesis test. Which z-scores would lead us to reject the null hypothesis with α = .05? I. z = -1.75 II. z = 1.75 III. z = -1.6
a. I only
Which of the following is a problem with using the z-score statistic? a. It requires knowing the population variance, which is often difficult to obtain. b. It requires very large samples in order to be effective. c. It is too cumbersome to calculate. d. It requires very small standard deviations in order to be effective.
a. It requires knowing the population variance, which is often difficult to obtain.
A treatment is administered to a sample selected from a population with a mean of μ = 40 and a standard deviation of σ = 6.25. After treatment, the sample mean is M = 45. Based on this information, the effect size as measured by Cohen's d can be classified as which of the following? a. Large effect b. Medium effect c. Small effect d. No effect
a. Large effect
In a normal sample distribution with n = 16, the null hypothesis is rejected. If the sample size is changed to 64 with all other factors staying the same, what happens to the z-score and the decision about the null hypothesis? a. The z-score is doubled, and the null hypothesis is still rejected. b. The z-score is multiplied by 6, and the null hypothesis is still rejected. c. The z-score is doubled, and we fail to reject the null hypothesis. d. The z-score is multiplied by 6, and we fail to reject the null hypothesis.
a. The z-score is doubled, and the null hypothesis is still rejected.
To evaluate the effect of a treatment, a sample is obtained from a population with a mean of μ = 25, and the treatment is administered to the individuals in the sample. After treatment, the sample mean is found to be M = 27.4 with SS = 64. If the sample consists of 9 individuals, what is the t statistic, and are the data sufficient to conclude that the treatment has a significant effect using a two-tailed test with α = .05?
a. t = 2.55, yes
A random sample is selected from a normal population with a mean of μ = 200 and a standard deviation of σ = 12. After a treatment is administered to the individuals in the sample, the sample mean is found to be M = 196. How large a sample is necessary for this sample mean to be statistically significant using a two-tailed test with α = .05? a. 30 b. 35 c. 40 d. 70
b. 35
If other factors are held constant, then how does the sample size affect the likelihood of rejecting the null hypothesis and the value for Cohen's d? a. A larger sample size increases the likelihood of rejecting the null hypothesis and increases the value of Cohen's d. b. A larger sample size increases the likelihood of rejecting the null hypothesis but does not change the value of Cohen's d. c. A larger standard deviation decreases the likelihood of rejecting the null hypothesis but increases the value of Cohen's d. d. A larger standard deviation decreases the likelihood of rejecting the null hypothesis and does not change the value of Cohen's d.
b. A larger sample size increases the likelihood of rejecting the null hypothesis but does not change the value of Cohen's d.
A random sample of n = 30 individuals is selected from a population with μ = 15, and a treatment is administered to each individual in the sample. After treatment, the sample mean is found to be M = 23.1 with SS = 400. In order to determine if the treatment had a significant effect, which of the following can we use? a. A z-score only. There is not enough information to use a t statistic. b. A t statistic. There is not enough information to use a z-score. c. Either a z-score or a t statistic. There is enough information for both. d. Neither a z-score nor a t statistic. There is not enough information to use either
b. A t statistic. There is not enough information to use a z-score
Assuming a normal distribution, which of the following would call for a one-tailed hypothesis test rather than a two-tailed test? a. Determining if attending Harvard influences IQ b. Determining if driving a red car increases the number of speeding tickets per year c. Determining if being male influences height d. Determining if being a teenager influences the number of hours of sleep per night
b. Determining if driving a red car increases the number of speeding tickets per year
A study is conducted to see if teenagers drive at faster average speeds than the general population of drivers. The average speed of the driving population is 35 mph. The null hypothesis is H0: μaverage driving speed of teenagers = 35 mph. What is the alternative hypothesis? a. H1: μdriving speed of teenagers = 35 mph b. H1: μdriving speed of teenagers ≠ 35 mph c. H1: μdriving speed of teenagers > 35 mph d. H1: μdriving speed of teenagers < 35 mph
b. H1: μdriving speed of teenagers ≠ 35 mph
Assuming all other factors stay the same, what happens to the proportion of the data in both tails as the degrees of freedom increases with a t statistic?
b. The proportion in the two tails combined decreases
What is the primary concern when selecting an alpha value? a. To make the null hypothesis easy to test b. To minimize Type I errors c. To minimize Type II errors d. To make z-scores easy to calculate
b. To minimize Type I errors
Which of the following is not an assumption for hypothesis tests with z-scores? a. random sampling b. small standard deviations c. normal sampling distribution d. independent observations
b. small standard deviations
Which of the following is not an advantage of a repeated-measures study over an independent-measures study? a. A repeated-measures design typically requires fewer subjects than an independent-measures design. b. A repeated-measures design is especially well suited for studying changes that take place over time. c. A repeated-measures design can be used on populations with high variances. d. A repeated-measures design reduces or eliminates problems caused by individual differences.
c. A repeated-measures design can be used on populations with high variances.
Which of the following explains why it is easier to reject the null hypothesis with a one-tailed test than with a two-tailed test with all the same parameters? a. Because the standard deviation in a one-tailed test is larger than that for a two-tailed test b. Because z-scores are calculated differently in a one-tailed test c. Because the critical region is all on one side in a one-tailed test and needs to be split between the two tails in a two-tailed test d. Because a two-tailed test uses a bimodal distribution
c. Because the critical region is all on one side in a one-tailed test and needs to be split between the two tails in a two-tailed test
If two different samples with different M's have the same z score, how can we find the pop. mean?
find average of two M's
Which of the following is an assumption for a related-samples t statistic? a. The observations within each treatment condition must be independent. b. The population distribution of the difference scores must be normal. c. Both a and b. d. Neither a nor b.
c. Both a and b.
Which of the following are correct ways of defining the power of a statistical test? I. The probability that the test will correctly reject a false null hypothesis II. The probability that the test will result in a Type II error III. The probability that the test will not result in a Type II error
c. I and II
Which of the following is not a step in a hypothesis test? a. State the null hypothesis about a population. b. Set the alpha level. c. If the sample data is not located in the critical region, we accept the null hypothesis. d. If the sample data is located in the critical region, we reject the null hypothesis.
c. If the sample data is not located in the critical region, we accept the null hypothesis.
A random sample is normally distributed. If all values in the sample and all values in the population are multiplied by 2, what is the impact on Cohen's d? a. Decreases b. Increases c. Stays the same d. It is impossible to tell
c. Stays the same If all values in the sample and all values in the population are multiplied by 2, then the sample mean and the population mean are multiplied by 2, which means the difference is also multiplied by 2. Further, the standard deviation is multiplied by 2. Since Cohen's d is the ratio of the mean difference to the standard deviation, there is no change in the value of Cohen's d.
A research report summarizes the results of the hypothesis test by stating, "z = 3.11, p < .01." Which of the following is a correct interpretation of this report? a. The null hypothesis was not rejected, and the probability of a Type I error is less than .01. b. The null hypothesis was not rejected, and the probability of a Type II error is less than .01. c. The null hypothesis was rejected, and the probability of a Type I error is less than .01. d. The null hypothesis was rejected, and the probability of a Type II error is less than .01.
c. The null hypothesis was rejected, and the probability of a Type I error is less than .01.
Which of the following is NOT an assumption for hypothesis testing using the t statistic? a. The population sampled must be normal. b. The values in the sample must consist of independent observations. c. The sample size must be greater than 30. d. The population standard deviation is unknown.
c. The sample size must be greater than 30.
what are the different values/effects for Cohen's d?
d = .20 →small effect d = .50 →medium effect d = .80 →large effect
A repeated-measures study with n = 26 participants produces a mean difference of MD = 3 points, SS = 500 for the difference scores, and t = 2.50. Calculate Cohen's d and r2 to measure the effect size for this study
d = 0.67, r2 = 0.2
What is the advantage of a repeated-subject research study? a. It uses exactly the same individuals in all treatment conditions. b. There is no risk that the participants in one treatment are substantially different from the participants in another c. A smaller number of subjects is required. d. All a, b, and c.
d. All a, b, and c.
In a matched-subjects design, how many variables can subjects be matched on? a. 1 b. 2 c. 3 d. All of the above
d. All of the above
What would be the result of setting an alpha level extremely small? a. There would be almost no risk of a Type I error. b. It would be very difficult to reject the null hypothesis. c. Neither a nor b d. Both a and b
d. Both a and b
A sample is selected from a population with μ = 50, and a treatment is administered to the sample. If the sample variance is s2 = 121, which set of sample characteristics has the greatest likelihood of rejecting the null hypothesis? a. M = 49 for a sample size of n = 15 b. M = 49 for a sample size of n = 75 c. M = 45 for a sample size of n = 15 d. M = 45 for a sample size of n = 75
d. M = 45 for a sample size of n = 75
When n is especially small, the t distribution is __________ and _______________
d. flatter, more spread out
If exactly 5% of the t distribution is located in the tail beyond t = 2.353, how many degrees of freedom are there?
df=3
Variance
equals the mean of the squared deviations the average squared distance from the mean
what is the relationship between standard error and sample size?
inverse relationship n↑=σx̅↓ n↓=σx̅↑
Which value is not included in the calculation of an estimated Cohen's d? a. μ b. n c. M d. s
n
A random sample is obtained from a population with μ = 120 and σ = 20, and a treatment is administered to the sample. Which of the following outcomes would be considered noticeably different from a typical sample that did not receive the treatment? a. n = 36 with M = 121 b. n = 36 with M = 123 c. n = 144 with M = 121 d. n = 144 with M = 124
n = 144 with M = 124
For samples selected from a population with μ = 90 and σ = 30, what sample size is necessary to make the standard distance between the sample mean and the population mean equal to 5 points?
n = 36
Formula for n using the standard error formula
n=(σ/σx̅)^2
why would it be easier to reject the null for one-tailed vs. two-tailed test?
one-tailed test= bigger critical region only on one side vs two-tailed test=critical regions split into smaller areas
r^2 (% variance/correlation coefficient squared)
r^2=(t^2)/(t^2+df)
For a sample of n = 16 scores with SS = 375, compute the sample variance and the estimated standard error for the sample mean.
s2 = 25, sM = 1.25
A repeated-measures study with a sample of n = 10 participants produces a mean difference of MD = 4.1 points with SS = 810 for the difference scores. For these data, find the variance for the difference scores and the estimated standard error for the sample mean
s^2 = 90, sMD= 3
what happens to a z score when a sample size is increased?
sample size=increased by factor x then z score increased by √x= z score increase i.e. if sample size increases by multiple of 4, then z score increases by √4=multiple of 2
what are the different effect sizes for r^2?
small effect= .01 medium effect= .09 large effect= .25
To evaluate the effect of a treatment, a sample is obtained from a population with a mean of μ = 31, and the treatment is administered to the individuals in the sample. After a treatment, the sample mean is found to be M = 32.7 with a sample variance of s2 = 4. If the sample size is n = 9, what is the t statistic, and is the data sufficient to conclude that the treatment increased the scores significantly? Use a one-tailed test and α = .01.
t = 2.55, which is not sufficient to reject the null hypothesis
For a repeated-measures study comparing two treatments with a sample of n = 16 participants, a researcher obtains a sample mean difference of MD = 3.3 with SS = 315 for the difference scores. Calculate the repeated-measures t statistic for these data using a two-tailed test, and determine if it is enough to reject the null hypothesis.
t = 2.88, the null hypothesis is not rejected
sM=√(s²/n)
t statistic (single sample) formula for standard error
what t statistic (of repeated measures study) would lead to rejecting null if α=.05 and one-tailed test?
t>1.708 OR t< -1.708
what t statistic would lead to rejecting null if α=.05 and one-tailed test?
t>1.812 OR t< -1.812
what t statistic (of repeated measures study) would lead to rejecting null if α=.05 and two-tailed test?
t>2.060 OR t< -2.060
what t statistic would lead to rejecting null if α=.05 and two-tailed test?
t>2.306 OR t< -2.306
what t statistic would lead to rejecting null if α=.01 and one-tailed test?
t>2.764 OR t< -2.764
what t statistic (of repeated measures study) would lead to rejecting null if α=.01 and one-tailed test?
t>2.896 OR t< -2.896
what t statistic would lead to rejecting null if α=.01 and two-tailed test?
t>2.947 OR t< -2.947
When computing z for a sample mean, which quantity is used?
the standard error
What happens to the variance when a sample size is increased?
variance decreases
df↑ =variance/standard error? how does this relate to the relationship between df of t statistic and critical regions?
variance/standard error↓ bc df ↑( denominator) recall: sd=√(SS/df) SO....sM/sxbar↓(denom) and t stat ↑=critical regions ↑ recall: t=(M-μ)/(sM/sxbar)
what is the formula for z score for a sample mean?
z= (M-μ)/σM
What is the interpretation of the z score formula?
z=(diff. b/w the data and hypothesis)/(standard distance b/w M and μ)
what z score would lead to rejecting null if α=.05 and one-tailed test?
z>1.65 OR z< -1.65
what z score would lead to rejecting null if α=.05 and two-tailed test?
z>1.96 OR z< -1.96
what z score would lead to rejecting null if α=.01 and two-tailed test?
z>2.58 OR z< -2.58
A researcher selects a sample of n = 25 individuals from a population with a mean of μ = 103 and administers a treatment to the sample. If the research predicts that the treatment will decrease scores, then what is the correct statement of the null hypothesis for a directional (one-tailed) test?
μ ≥ 103
how is standard error affected if sample size is multiplied by a number?
σx̅ will be divided by half the original multiplying number opposite function (multiplying/dividing) and half number
What is the formula for standard error?
σx̅=σ/√n