chapter 7 stats
What is the P-value of the test in this situation?
0.01 < P < 0.02
Suppose the researcher had wished to test the following hypotheses: H0 : μ1 = μ2, Ha: μ1 > μ2 The numerical value of the two-sample t statistic is
1.20.
The numerical value of the standard error of the difference in sample means is
1.28.
The degrees of freedom for these t procedures are closest to
14
The cholesterol levels (in milligrams per liter) of three randomly selected adult males is 260, 315, and 295. The standard error of the mean of these three levels is
16.07
From previous work, the researchers expected the population of responses to be Normal. The insects are a random sample from a cockroach population grown in the laboratory. A 95% confidence interval for the mean amount of d-glucose in cockroach hindguts under these conditions is
18.69 to 70.19
The degrees of freedom for these t procedures are closest to
183
What is a 95% confidence interval for μ1 - μ2? (Do not assume equal population variances.)
2 ± .99 hours
A 95% confidence interval for μ1 - μ2 is
2 ±1.07 m
We are interested in testing the following hypotheses: H0 : μM = μF, Ha : μM ≠ μF From the 95% confidence interval computed in the previous question, we
can reject the null hypothesis at significance level alpha 5%
Is the difference between the two population means, μ1 - μ2, included in the 95% confidence interval?
can't tell
It is not unreasonable to assume that the population standard deviations, although unknown in value, are equal, i.e., . Using the summary statistics in the table above, what would be the pooled estimate of the common standard deviation σ?
d.
What are the appropriate hypotheses to use in this case to determine if the herbal supplement is effective in achieving weight loss in obese males?
e. H0: = 0 and Ha: > 0
At significance level 5%, we
fail to reject the null hypothesis.
The data are statistically significant at the 1% level of significance.
false
A 99% confidence interval for μ1 - μ2 is
4 ± 9.8 yards.
If the 98% confidence interval were determined to be (45.2, 49.6) an interval, which may or may not be correct, what would the researchers conclude?
At the 0.02 level of significance, there is evidence to conclude that the sample comes from a population with a mean different from μ = 45
Suppose we were not sure if the distribution of weights was Normal. In which of the following circumstances would we not be safe using a t procedure in this problem?
A stemplot of the data has a large outlier.
What is a 90% confidence interval for μ, the mean change in SAT-M score?
(-22.56, 72.56)
What is a 95% confidence interval for μ2 - μ1? Assume the population standard deviations are equal.
(-4.57, 24.57)
Scores on the SAT Mathematics test are believed to be Normally distributed. The scores of a simple random sample of five students who recently took the exam are 550, 620, 710, 520, and 480. What is a 95% confidence interval for μ, the population mean score on the SAT Math test?
(463.4, 688.6)
A 90% confidence interval for μ1 - μ2 is
-0.6 ± 0.41
What is a 90% confidence interval for difference between the two population means?
-1 ± 1.74 years
We test the hypotheses H0 : μ1 = μ2, Ha : μ1 ≠ μ2. The numerical value of the two-sample t statistic is
-2.40.
Based on the data, the value of the one-sample t statistic is between 0.10 and 0.05.
-2.89.
The height (in inches) of males in the United States is believed to be Normally distributed with mean μ. The average height of a random sample of 50 American adult males is = 69.72 inches, and the standard deviation of the 50 heights is s = 4.23. The standard error of is
0.598.
What is the approximate P-value for the 2 sided test?
Between 0.0025 and 0.005
Suppose the mean and standard deviation obtained were based on a sample of size n = 25 postal workers rather than 100. What do we know about the value of the P-value?
It would be larger
Which of the following is not a requirement for using the matched-pairs t procedure?
Measurements are taken on completely independent subjects.
Suppose the researcher had wished to test the hypotheses H0: μ1 = μ2 versus Ha: μ1 ≠ μ2. What can we say about the value of the P-value? Do not assume the variances of the two populations are equal
P-value < 0.01
Suppose the researcher had wished to test the hypotheses H0: μ1 = μ2 versus Ha: μ1 ≠ μ2. What can we say about the value of the P-value? Do not assume the variances of the two populations are equal.
P-value < 0.01
Suppose we wish to test if the mean amount of damage equal to $30,000 or not. Based on the previously calculated confidence interval, what would we conclude at the .05 significance level?
There is insufficient evidence to conclude the mean amount of damage in the population is $30,000.
Which of the following is correct?
Two-sample procedures can be used with random samples from each group and for experiments where a single population is randomized to the two groups.
Which of the following is correct?
We are 95% confident that each boy has an IQ score that is 5 points greater than their matched girl.
Suppose we do not believe that students tend to improve their SAT-M score the second time they take the test. Based on the 90% confidence interval previously calculated (-22.56, 72.56), we wish to test H0: μ = 0 versus Ha: μ ≠ 0 at the 5% significance level. Determine which of the following statements is true:
We cannot reject H0 since the value 0 falls in the 90% confidence interval and would therefore also fall in the 95% confidence interval
When performing a t procedure, it is best to look at the data first and then determine if you should do a one-sided or two-sided test.
false
When planning a two sample design, it is best to use a different sample size for each sample.
false
When the underlying assumptions for the use of t procedures are not exactly met, the probability calculations are still valid as long as the data are very close to Normal.
false
Determine whether each of the following statements is true or false.
false true true
Two sample procedures are less robust when the sample sizes are not similar.
true
When the sample size is very large, the corresponding t distribution is very close to the normal distribution.
true
Determine whether each of the following statements is true or false.
true false true
What type of inference procedure is appropriate here?
Matched pairs t procedure
The rationale for avoiding the pooled procedures for inference is that
all of the above
The standard error of the mean is
2.68
If we were to use the pooled t test, what would be the degrees of freedom?
25
An SRS of 20 third grade children is selected in Chicago and each is given a test to measure their reading ability. We are interested in a 90% confidence interval for the population mean score. In the sample, the mean score is 64 points and the standard deviation is 12 points. The margin of error associated with the confidence interval is
4.64 points
Based on these data, what is the 98% confidence interval estimate for the mean oxygen uptake for the population from which the sample of students was selected?
47.4 ± 2.457 (5.3/31)
What are the appropriate degrees of freedom for this test?
99
When using a software package (e.g., SPSS) to compute t procedures and only the results for the two-sided alternative hypothesis are given, what can you do to obtain the P-values for the one-sided alternative hypothesis?
Divide the P-value by 2.
The manager of an automobile dealership is considering a new bonus plan to increase sales. Currently, the mean sales rate per salesperson is five automobiles per month. The correct set of hypotheses to test the effect of the bonus plan is
H0: μ = 5, Ha: μ > 5
What would be the hypotheses for this problem?
H0: μ1 = μ2 versus Ha: μ1 ≠ μ2
What is the appropriate alternative hypothesis in this situation?
Ha: μ ≠ 45
Which of the following would lead us to believe that the t procedures were not safe to use here?
The market analysts deemed that the volunteers could not be considered a simple random sample from the population.
The method used to compute this confidence interval has a 95% probability of producing an interval that captures
The method used to compute this confidence interval has a 95% probability of producing an interval that captures
When can the pooled two-sample t procedure be used?
When two simple random samples are taken from two populations with the same standard deviation.
To which of the following would it have been most important that the researchers be blind during the experiment?
Which diet the cows received
You are thinking of using a t procedure to construct a 95% confidence interval for the mean of a population. You suspect the distribution of the population is not Normal and may be skewed. Which of the following statements is correct?
You may use the t procedure provided your sample size is large, say at least 50
A pharmacist notices that a majority of his customers purchase a certain name brand medication rather than the generic—even though the generic has the exact same chemical formula. To determine if there is evidence that the name brand is more effective than the generic, he talks with several of his pharmaceutical colleagues, who agree to take each drug for two weeks, in a random order, in such a way that neither the subject nor the pharmacist knows what drug they are taking. At the end of each two-week period, the pharmacist measures their gastric acid levels as a response. The proper analysis is to use
a matched pairs t test.
Below are normal quantile plots for two data sets. For which set(s) is(are) inference using the t distribution valid?
b.
The P-value for the one-sample t test is
below 0.01
A researcher wishes to compare the effect of two stepping heights (low and high) on heartrate in a step-aerobics workout. A collection of 50 adult volunteers was randomly divided into two groups of 25 subjects each. Group 1 did a standard step-aerobics workout at the low height. The mean heartrate at the end of the workout group 1 was 1 = 90.00 beats per minute with a standard deviation s1 = 9 beats per minute. Group 2 did the same workout but at the high step height. The mean heart-rate at the end of the workout for the subjects in group 2 was 2 = 95.08 beats per minute with a standard deviation s2 = 12 beats per minute. Assume the two groups are independent and the data are approximately Normal. Let µ1 and µ2 represent the mean heart rates we would observe for the entire population if all members of this population did the workout using the low or high step height, respectively. Since the sample standard deviations are similar, it is safe to assume that the population standard deviations are equal. Suppose the researcher had wished to test the hypotheses H0: µ1 = µ2, H a: µ1 < µ2. The P-value for the test is
between 0.05 and 0.01
We wish to see if the dial temperature for a certain model oven is properly calibrated. Four ovens of a certain model are selected at random. The dial on each is set to 300° F and after one hour, the actual temperature of each is measured. The temperatures measured are 305°, 310°, 300°, and 305°. Assuming that the actual temperatures for this model when the dial is set to 300° are Normally distributed with mean µ, we test whether the oven is properly calibrated by testing the hypotheses H0: µ = 300, Ha: µ ≠ 300. Based on the data, the P-value for this test is
between 0.10 and 0.05.
The researcher tests the following hypotheses: H0 : μ1 = μ2, Ha : μ1 ≠ μ2 The 90% confidence interval is 2 ± 0.83 meters. Based on this confidence interval,
both a and b are correct
Data on the blood cholesterol levels of 34 rats (milligrams per deciliter of blood) give = 85 and s = 12. A 95% confidence interval for the mean blood cholesterol of rats under this condition is computed. Which of the following would be most worrisome for the validity of the confidence interval?
the presence of a clear outlier in the raw data
Matched pairs t procedures are for use on subjects that are _______.
the same or similar
A researcher wants to know if calcium is an effective treatment for lowering blood pressure. He assigns one randomly chosen group of subjects to take calcium supplements; the other group will get placebo. At the end of the treatment period, he measures the difference in blood pressure. The 50 members of the calcium group had blood pressure lowered by an average of 25 points with standard deviation 10 points. The 50 members of the placebo group had blood pressure lowered by an average of 15 points with standard deviation 8 points. To analyze this information we will use a
two-sample t procedure.
If, in fact, the true average distance at which drivers are able to read the sign is 450 feet, what type of error would the researcher have committed?
type 1 error
What is the value of the test statistic?
-2.18
The researcher tests the following hypotheses: H0 : μ1 = μ2, Ha : μ1 ≠ μ2 The P-value for the test is
.0004
two professors find the following information: = 111, = 120, s1 = 7, and s2 = 11. Suppose the professors had wished to test the hypotheses H0: μ1 = μ2 versus Ha: μ1 < μ2. What can we say about the value of the P-value?
0.01 < P-value < 0.025
Suppose we were to use the unpooled t test. The t statistic for comparing the mean prices is 1.9. What can we say about the value of the P-value?
0.05 < P-value < 0.10
I took a sample of the grade point averages for students in my class. For 25 students, the standard deviation of grade points was 0.65 and the mean was 2.89. The standard error for the sample was
0.13
The test degrees of freedom are roughly 13. The P-value for the test is closest to
0.3.
How different are male and female zebra finches in bill color? The margin of error for a 95% confidence interval for the difference in mean bill color μF - μM (in hue degree) is
0.74.
The heights (in inches) of males in the U.S. are believed to be Normally distributed. The average height of a random sample of 25 American adult males is found to be = 69.72 inches with a standard deviation of s = 4.15. What is the standard error of ?
0.83
The P-value for the test is closest to
2%
A researcher wanted to know if the content of television programs has an impact on viewers' recall of ad content. He randomly assigned 18 people to view a program with violent content, and 20 people to view a program with neutral content. The same 9 commercials were inserted into each program. After viewing the program, the subjects were asked to recall the brands advertised in the commercials. The average number of brands recalled for those who saw the violent program was 3.77 with standard deviation 1.87. The average number of brands recalled for those who saw the neutral program was 4.65 with standard deviation 1.67. The t* multiplier (rounded to 3 decimal places) for a 95% confidence interval using the pooled t-interval is
2.028.
We wish to see if the dial indicating the oven temperature for a certain model oven is properly calibrated. Four ovens of this model are selected at random. The dial on each is set to 300°F, and after one hour, the actual temperature of each is measured. The temperatures measured are 305°, 310°, 300°, and 305°. Assume that the distribution of the actual temperatures for this model when the dial is set to 300° is Normal. To test if the dial is properly calibrated, we will test if the actual (population) mean temperature is something other than 300°. Based on the data, what is the value of the t statistic?
2.45
A simple random sample of five female basketball players is selected. Their heights (in cm) are 170, 175, 169, 183, and 177. What is the standard error of the mean of these height measurements?
2.538
A 90% confidence interval for μ is
25.0 ± 47.54
Suppose we wish to determine if there tends to be a difference in height for the seedlings treated with the different herbicides. To answer this question, we decide to test the hypotheses H0: μ2 - μ1 = 0, Ha: μ2 - μ1 ≠ 0. What is the value of the two-sample t statistic?
3.43
The researcher tests the following hypotheses: H0 : μ1 = μ2, Ha : μ1 ≠ μ2 The numerical value of the two-sample t statistic is
3.700
Assuming two-sample t procedures are safe to use, what is a 99% confidence interval for ? (Assume the population standard deviations are equal.)
4 ± 9.76 yards
Post-exposure radiation levels are approximately Normally distributed with mean μ. The levels (in Sv) of a random sample of three trauma victims who were recently exposed are 5.5, 6.2, and 4.8. A 95% confidence interval for μ based on these data is
5.50 ± 1.74
A random sample of 20 observations produced a sample mean = 92.4 and s = 25.8. What is the value of the standard error of ?
5.8
You want to know how a new brand of diet soup will fare among customers, especially compared to a classic brand. Which of the following designs is an example of a matched pairs design?
50 subjects taste and rate the 2 brands of diet soup in unmarked cups. The order of the soup type given is randomized.
A simple random sample of six male patients over the age of 65 is being used in a blood pressure study. The standard error of the mean blood pressure of these six men was 22.8. What is the standard deviation of these six blood pressure measurements?
55.85
What is a 95% confidence interval for μ, the population mean time the postal service employees have spent with the postal service?
7 ± 0.4
A 95% confidence interval for μ1 - μ2, based on two independent samples of sizes 18 and 20, respectively, gives us (45.6, 56.7). What would be the margin of error for a 99% confidence interval for μ1 - μ2? Assume the standard deviations of the two populations are equal.
7.44
To estimate μ, the mean salary of full professors at American colleges and universities, you obtain the salaries of a random sample of 400 full professors. The sample mean is = $73,220, and the sample standard deviation is s = $4400. A 99% confidence interval for μ is
73,220 ± 569 (t was used).
When calculating a one-sample t confidence interval, which level of confidence will give the smallest interval?
80%
When calculating a two-sample t confidence interval, which level of confidence will give the smallest interval?
80%
A researcher wished to compare the average amount of time spent in extracurricular activities by high school students in a suburban school district with that in a school district of a large city. The researcher obtained an SRS of 60 high school students in a large suburban school district and found the mean time spent in extracurricular activities per week to be = 6 hours with a standard deviation s1 = 3 hours. The researcher also obtained an independent SRS of 40 high school students in a large city school district and found the mean time spent in extracurricular activities per week to be = 4 hours with a standard deviation s2 = 2 hours. Let µ1 and µ2 represent the mean amount of time spent in extracurricular activities per week by the populations of all high school students in the suburban and city school districts, respectively. If the researcher used the more accurate software approximation to the degrees of freedom, he would have used which of the following for the number of degrees of freedom for the two-sample t procedures?
98
The degrees of freedom for these t procedures are closest to
98
A 95% confidence interval for the difference in average household income in the two neighboring towns is (4.5721, 9.8279). What can be said (with certainty) about the value of the P-value for testing the hypotheses H0: μ1 = μ2 versus Ha: μ1 ≠ μ2?
< 0.05 < 0.10
Suppose we were not sure if the distribution of net weights were Normal. In which of the following circumstances would we not be safe using a t procedure in this problem?
A boxplot of the data shows the presence of a large outlier.
Which of the following statements about the two-sample t procedures for significance tests and confidence intervals for the difference of the means, , is (are) True?
Although Normality is a requirement, with large sample sizes the central limit theorem guarantees that the results will be approximately correct for other population distributions The two samples must be independent The two samples must be simple random samples.
Assume that sample data, based on two independent samples of size 25, give us = 505, = 515, s1 = 23, and s2 = 28. The two-sample 95% confidence interval is found to be (-24.57, 4.57). Select the statement(s) that is (are) TRUE.
Based on the confidence interval, we cannot conclude at the 5% significance level that there is a difference between the two population means, μ2 and μ1. The margin of error for the difference between the two sample means would be smaller if we were to take larger samples. If a 99% confidence interval were calculated instead of the 95% interval, it would include more values for the difference between the two population means.
A 90% confidence interval for the difference in mean heights for the two herbicides is (0.2, 14.6). Which statement is correct?
Both choices are incorrect.
The null and alternative hypotheses to test whether there is significant evidence of a different mean adaptation time (in steps) for the two disturbances are:
H0 : μDifference A-B = 0, Ha : μDifference A-B ≠ 0
Determine which of the following statements regarding the decision the representative would make is true.
He would reject H0 at a significance level of 0.05 but not at 0.025.
Which of the following is correct?
If we were to repeat the experiment many times, then the confidence interval will contain the population value about 95% of the time.
We are interested in testing a null hypothesis about a population mean μ being equal to a specified value using a simple random sample of size 25. In the past the population variable has shown a slight tendency to non-Normality (slight skewness but no strong outliers). Why can we safely use a t procedure in this testing situation?
Only B and C are justifications for using the t procedure here.
Suppose the political analyst had wished to test the hypotheses H0: μ1 = μ2 versus Ha: μ1 < μ2. What can we say about the value of the P-value?
P-value > 0.10
What can you say about the value of the P-value?
P-value > 0.10
Which of the following would lead us to believe that the t procedures were not safe to use in a two-sample problem with 100 subjects in each group?
Several outliers were detected in each group.
What assumption must be true for this test result to be correct?
The population distributions must be roughly Normal
If you use the 0.05 level of significance, what conclusion would you reach?
There is insufficient evidence to conclude the mean yellowness of the two types of feathers are different. There is insufficient evidence to conclude the mean yellowness of the two types of feathers are the same.
A highway safety researcher is studying the average maximum distance at which drivers are able to read a highway traffic sign. The designer of the sign claims it will be over 450 feet. Sixteen randomly selected drivers are asked at what distance they can still read the sign. The average maximum distance (in feet) of these 16 drivers is 498 feet with a standard deviation of 95 feet. The highway safety researcher wishes to use these data to test if the mean maximum distance at which drivers are able to read the sign is greater than 450 feet at a significance level of α = 0.05. What is the conclusion to the test?
There is sufficient evidence to conclude that the population mean maximum distance at which drivers are able to read the sign is greater than 450 feet.
A bank is investigating ways to entice customers to charge more on their credit cards. (Banks earn a fee from the merchant on each purchase, and hope to collect interest from the customers as well). A bank selects a random group of customers who are told their "cash back" will increase from 1% to 2% for all charges above a certain dollar amount each month. Of the 500 customers who were told the increase applied to charges above $1000 each month, the average increase in spending was $527 with standard deviation $225. Of the 500 customers who were told the increase applied to charges above $2000 each month, the average increase in spending was $439 with standard deviation $189. When testing whether or not the increases in spending are different, the test is significant at (check all that apply or none)
all three choices
The data represent a random sample of T-cells grown in lab conditions. Computing a 95% for the mean T-cell velocity μ is
appropriate despite the clearly skewed data, because the sample size is large enough
The P-value for the appropriate test to detect a difference in the two disturbances is
between 0.025 and 0.05
The air in poultry-processing plants often contains fungus spores, especially in the summer. Inadequate ventilation can affect the health of the workers. To measure the presence of spores, air samples are pumped to an agar plate and "colony forming units (CFUs)" are counted after an incubation period. An inspector for the Occupational Safety and Health Administration collects air samples from both the kill room and the processing room of a large poultry-processing plant on a random sample of 4 days over one summer. The data collected are matched pairs data, because
each day one measurement was taken from both rooms.
One sample t test procedures are for use on subjects that are _______.
independant
When performing t procedures on data with small sample sizes, the margin of error will likely be _____.
large
A medical researcher wishes to investigate the effectiveness of exercise versus diet in losing weight. Two groups of 25 overweight adult subjects are used, with a subject in each group matched to a similar subject in the other group on the basis of a number of physiological variables. One group is placed on a regular program of vigorous exercise, but with no restriction on diet, and the other group is placed on a strict diet, but with no requirement to exercise. The weight losses after 20 weeks are determined for each subject, and the difference between matched pairs of subjects (weight loss of subject in exercise group - weight loss of matched subject in diet group) is computed. The mean of these differences in weight loss is found to be -2 pounds with standard deviation s = 6 pounds. Is this evidence of a difference in mean weight loss for the two methods? To test this, consider the population of differences (weight loss overweight adult would experience after 20 weeks on the exercise program) - (weight loss the same adult would experience after 20 weeks on the strict diet). Let μ be the mean of this population of differences and assume their distribution is approximately Normal. We test the hypotheses H0 : μ = 0 versus Ha : μ ≠ 0, using the matched pairs t test. The P-value for this test is
larger than 0.10
Suppose the researcher had wished to test the following hypotheses: H0: μ1 = μ2, Ha: μ1 > μ2 The P-value for the test is
larger than 0.10.
The P-value for the test is
larger than 0.10.
Suppose a simple random sample size of n is drawn from an appropriately normal population. What degrees of freedom should be used to perform a one sample t procedure?
n-1
As the degrees of freedom become larger, the difference between the t and z distributions becomes
narrower
Is the probability that the difference between the two population means, μ1 - μ2, falls between 45.6 and 56.7 equal to 0.95?
no
A 95% confidence interval for the difference between population means, based on two independent samples of sizes 18 and 20, respectively, gives us (45.6, 56.7).
no cant tell yes
It has been claimed that women live longer than men; however, men tend to be older than their wives. Ages of sixteen husbands and wives from England were obtained. These data should be analyzed with a
paired samples t test.
Two sample t procedures are used when _____.
subjects in one sample are completely unrelated to the subjects in the other sample
The value of the test statistic for the test is
t = 0.975
What is the value of the t test statistic?
t = 1.05
The test statistic for the appropriate test of hypothesis is
t = 2.53.
With α = 0.05, what observed values of the test statistic would cause the sample results to be declared statistically significant?
t > 2.015
When testing the hypothesis of equal means of two independent random samples, which distribution is used?
t distribution
If we were to test H0: against Ha: , what would be the value of the test statistic (assuming equal population standard deviations)?
t(40) = 3.46
A local farmer is interested in comparing the yields of two varieties of tomatoes. In an experimental field, she selects 40 locations and assigns 20 plants from each variety at random to the locations. She determines the average per plant (in pounds). She computes a 95% confidence interval for the difference in mean yields between the two varieties using the two-sample t procedures with the resulting interval (2.13, 6.41). For testing using the two-sample t procedures we can say that
the P-value must be less than 0.05.
At the 5% significance level we can conclude that the merged firms have a smaller price-earnings ratio, on average, than the non-merged firms.
true
Robustness refers to how sensitive probability calculations are when assumptions are not met.
true
To determine if the diet leads to weight loss, we test the following hypotheses.
we would not reject H0 at significance level 0.10.
has the value t = 1.93. Based on this information
we would reject the null hypothesis at α = 0.10.
A 95% confidence interval for the difference in average household income in the two neighboring towns is (4.5721, 9.8279). At the 5% level of significance, should the group of students reject the null hypothesis?
yes
Is the difference between the two sample means, - , included in the 95% confidence interval?
yes
We would like to know the average amount of time students spend studying for their classes each week. A random sample of 100 students are surveyed. They reported spending an average of 7.6 hours studying, with s = 1.2 hours. Based on these data, find a 95% confidence interval for the average amount of time a student studies for classes each week.
(7.36, 7.84)
