Deck 10: Inference From Small Samples

Question 1

In testing vs. the critical value is determined from the F-distribution table with an upper tail area equal to half the value of the level of significance.

Answer 1

True

Question 2

In testing vs. the null hypothesis will be rejected if the ratio is substantially longer than 1.0.

Answer 2

True

Question 3

In testing vs. the F-test statistic is calculated as F = .

Answer 3

False

Question 4

In testing vs. if and then the calculated value of the test statistic is F = 1.60.

Question 5

If two population variances have been tested and found to be equal, then it is reasonable to conclude that the two random samples selected from the two populations have equal variances.

Question 6

A two-tailed test for two population variances could have the null hypothesis to be written as .

Question 7

In testing for the equality of two population variances, when the populations are normally distributed, the 5% level of significance has been used. To determine the rejection region, it will be necessary to refer to the F table corresponding to an upper-tail area of 0.05.

Question 8

The value of F that locates an area 0.01 in the upper tail of the F-distribution for = 15 and = 10 is 3.80.

Question 9

The F-distribution is used in testing vs. .

Question 10

We can use either the z-test or the t-test to determine whether two population variances are equal.

Question 11

The test statistic employed to test is , which is F distributed with degrees of freedom, provided that the two populations are F distributed.

Question 12

We can use either the z-test or the t-test to determine whether two population variances are equal.

Question 13

When comparing two population variances, we use the ratio rather than the difference .

Question 14

When the necessary conditions are met, a two-tail test is being conducted at = 0.05 to test . The two sample variances are , and the sample sizes are . The calculated value of the test statistic will be F = 2.

Question 15

When the necessary conditions are met, a two-tail test is being conducted at = 0.05 to test . The two sample variances are , and the sample sizes are . The rejection region is F > 2.16 or F < 0.4347

Question 16

In testing the equality of two population variances, when the populations are normally distributed, the 5% level of significance has been used. To determine the rejection region, you will refer to the F table corresponding to an upper-tail area of 0.025.

Question 17

The value of F with area 0.05 to its right for = 6 and = 9 is 3.37.

Question 18

In testing vs. the larger the sample sizes from the two populations, the smaller will be the chance of committing a Type I error.

Question 19

In a one-tailed test of the equality of two population variances, the F-test statistic is calculated by placing the larger sample variance in the numerator.

Question 20

The F-test used for testing the difference in two population variances is always a one-tailed test.

Question 21

The test for the equality of two population variances assumes that each of the two populations is normally distributed.

Question 22

If two random samples of 10 and 12 observations produced sample variances equal to 7.50 and 3.20, respectively, then the calculated value of the test statistic when testing vs. is equal to:

A) 7.50
B) 3.20
C) 10.70
D) 2.34
E) 7.82

Question 23

In testing vs. , which of the following statements is true?

A) The larger sample variance must be used as the numerator of the F-test statistic.
B) The level of significance needs to be doubled before finding the critical value in the F-table.
C) The alternative hypothesis must contain the equality.
D) You are free to decide which of the two populations you want to call "Population 1."
E) The level of significance needs to be doubled before finding the critical value in the F-table and the alternative hypothesis must contain the equality.

Question 24

When testing for the difference between two population variances with sample sizes of and , the number of degrees of freedom is:

A) 8 and 10
B) 7 and 9
C) 2 and 18
D) 18 and 2
E) 9 and 11

Question 25

When testing vs. two random samples of sizes 10 and 8, respectively, are used. The calculated value of the test statistic is found to be equal to 2.22. Which of the following statements is true?

A) H₀ must be rejected at α = .05 .
B) H₀ cannot be rejected at α = .05 .
C) There is sufficient evidence at α = .05 to conclude that a difference in the population variances exist.
D) H₀ cannot be rejected at α = .10 .
E) None of these.

Question 26

The F-distribution is symmetric.

Question 27

The sampling distribution of the ratio of two sample variances is said to be F distributed provided that:

A) the samples are independent
B) the populations are normal with equal variances
C) the samples are dependent and their sizes are large
D) the samples are independently drawn from two normal populations
E) none of these

Question 28

A stylist realizes that the quicker she can give a permanent, the happier the customer will be. She has made a slight adjustment to her traditional routine in hopes of reducing the time required. A random sample of 13 permanents from before and 16 after were timed. The results of the tests are listed below. Assume the two samples were taken from normal populations with equal variances. Do the sample variances present sufficient evidence to indicate that the population variance before is greater than the variance after? Test the appropriate hypotheses using a significance level of 0.05. Interpret your results.
Test Statistic = ______________
Reject Region: Reject H₀ if F > ______________
Conclusion: ______________
There is ______________ to indicate that the population variance before is greater than the variance after.
Construct a 95% confidence interval (CI) for .
CI = ______________ Enter (n1, n2)
Find the approximate p-value for the test and interpret its value.
______________ Enter (n1, n2)

Question 29

Which of the following distributions is used in testing vs.

A) Chi-square distribution
B) Normal distribution
C) F-distribution
D) t-distribution
E) Chi-square distribution and Normal distribution

Question 30

The sampling distribution of the ratio of two independent sample variances selected randomly from normal populations with equal variances is:

A) a normal distribution
B) a t distribution
C) an F distribution
D) a binomial distribution
E) a chi-square distribution

Question 31

Which of the following statements is false for an F-distribution?

A) Variables that are F distributed range from 0 to.
B) Exact shape of the distribution is determined by two numbers of degrees of freedom.
C) Degrees of freedom for the denominator are always smaller than the degrees of freedom for the numerator.
D) Degrees of freedom for the numerator can be larger, smaller, or equal to the degrees of freedom for the denominator.
E) Degrees of freedom for the denominator are always smaller than the degrees of freedom for the numerator and degrees of freedom for the numerator can be larger, smaller, or equal to the degrees of freedom for the denominator.

Question 32

Which of the following is not a property of the F distribution?

A) F-distributions are nonsymmetrical.
B) F-distributions have (n - 1) degrees of freedom.
C) F can assume only positive values.
D) There are many F-distributions and each has a different shape.
E) F-distributions are nonsymmetrical and F can assume only positive values.

Question 33

Two soft drink machines dispense liquids of 10 ounces, on the average. The question is whether the two machines are equally consistent (i.e., equally variable) in the dispensing of the liquid. To answer this question, a sample of size 10 was obtained from each machine and the sample standard deviations were computed to be = 1.87 ounces and = 1.25 ounces. Perform the appropriate test for equality of variances using = 0.05.
Test Statistic = ______________
Reject Region: Reject H₀ if F> ______________
Conclusion: ______________
One ______________ conclude that there is a significant difference in the variability of liquid dispensed between the two machines.

Question 34

In constructing a 90% interval estimate for the ratio of two population variances, , two independent samples of sizes 40 and 60 are drawn from the populations. If the sample variances are 515 and 920, then the lower confidence limit is:

A) 0.352
B) 0.341
C) 0.890
D) 0.918
E) 0.245

Question 35

The test for the equality of two population variances is based on the:

A) difference between the two sample variances
B) ratio of the two sample variances
C) difference between the two population variances
D) difference between the two sample variances divided by the difference between the two population variances
E) difference between the two population variances divided by the two sample variances

Question 36

The F-distribution is the sampling distribution of the ratio of:

A) two normal population variances
B) two normal population means
C) two sample variances provided that the samples are independently drawn from two normal populations with equal variances
D) two sample variances provided that the sample sizes are large
E) none of these

Question 37

The assumption(s) for to have an F-distribution is (are):

A) random samples are drawn from each of two normal populations
B) the samples are independent
C) the variability of the measurements in the two populations is the same and can be measured by a common variance,
D) all of these are required assumptions
E) none of these

Question 38

In testing for the equality of two population variances, when the populations are normally distributed, the 10% level of significance has been used. To determine the rejection region, it will be necessary to refer to the F table corresponding to an upper-tail area of:

A) 0.90
B) 0.05
C) 0.20
D) 0.10
E) 0.85

Question 39

The statistical distribution used for testing the difference between two population variances is the:

A) Student t-distribution
B) standard normal distribution
C) binomial distribution
D) F-distribution
E) chi-square distribution

Question 40

For an F-distribution, the number of degrees of freedom for the denominator:

A) must be larger than the number of degrees of freedom for the numerator
B) must be equal to the number of degrees of freedom for the numerator
C) must be smaller than the number of degrees of freedom for the numerator
D) can be larger, smaller, or equal to the number of degrees of freedom for the numerator
E) none of these

Question 41

The average total SAT scores (verbal plus math) were recorded for two groups of students: one group planning to major in engineering and one group planning to major in language/literature. To use the two-sample t test with a pooled estimate of you must assume that the two population variances are equal. Test this assumption using the F test for equality of variances using = 0.05.
Test Statistic = ______________
Reject Region: Reject H₀ if F > ______________
Conclusion: ______________
One ______________ conclude that the variances are different.
What is the approximate p-value for the test?
______________

Question 42

An experimenter is concerned that variability of responses using two different experimental procedures may not be the same. He randomly selects two samples of 16 and 14 responses from two normal populations and gets the statistics: s₁² = 55, and s₁² = 118, respectively. Do the sample variances provide enough evidence at the 10% significance level to infer that the two population variances differ?
Test statistic = ______________
Critical Value(s) = ______________
Conclusion: ______________
Interpretation: __________________________________________
Estimate with 90% confidence the ratio of the two population variances.
______________

Question 43

The 90^th percentile of a chi-squared distribution with 15 degrees of freedom is equal to 22.3072.

Question 44

The chi-squared distribution is:

A) symmetric around zero
B) positively skewed
C) negatively skewed
D) mound-shaped
E) positively skewed and mound-shaped

Question 45

The chi-square distribution is skewed to the left (negatively skewed), but as degrees of freedom increase, it approaches the shape of the binomial distribution.

Question 46

The test statistic used to test hypotheses about the population variance is , which is chi-squared distributed with n -1 degrees of freedom when the population is normally distributed with variance equal to .

Question 47

The t-distribution with n - 1 degrees of freedom is used when testing a null hypothesis for a population variance.

Question 48

For a given level of significance, increasing the sample size will tend to increase the chi-square critical value used in testing the null hypothesis about a population variance.

Question 49

The area under a chi-squared curve with 10 degrees of freedom, which is captured between the critical values is .

Question 50

Independent random samples from two normal populations produced the variances listed here: Do the data provide sufficient evidence to indicate that differs from ? Test using = .05.
Test Statistic = ______________
Reject Region: Reject H₀ if F> ______________
Conclusion: ______________
One ______________ conclude that the variances are different.
Find the approximate p-value for the test and interpret its value.
______________ Enter (n1, n2)
Develop a 95% confidence interval for .
CI = ______________ Enter (n1, n2)

Question 51

If you wish to test vs. at the .05 level of significance using a sample of 15 observations, the critical value to be used is 23.685.

Question 52

The area to the right of a chi-squared variable is 0.025. For 5 degrees of freedom, the critical value is 11.143.

Question 53

A food processor wants to compare two preservatives for their effects on retarding spoilage. Suppose 16 cuts of fresh meat are treated with preservative A and 16 are treated with preservative B, and the number of hours until spoilage begins is recorded for each of the 32 cuts of meat. The results are summarized in the table below: Calculate the value of the test statistic for testing the equality of the population variances, and write the proper conclusion for = 0.05.
Test statistic = ______________
Critical Value(s) = ______________
Conclusion: ______________
Interpretation: __________________________________________
Develop the 95% confidence interval estimate of the ratio of the two population variances.
______________

Question 54

A statistician wants to test for the equality of means in two independent samples drawn from normal populations of people enrolled in a diet program. However, he will not perform the equal-variance t-test of the difference between the population means if the condition necessary for its use is not satisfied. The number of pound lost at the completion of the program data follow: Can the statistician conclude at the 5% significance level that the required condition is not satisfied?
Test statistic = ______________
Critical Value(s) = ______________
Conclusion: ______________
Interpretation: __________________________________________
Estimate with 95% confidence the ratio of the two population variances.
______________

Question 55

A right-tailed area in the chi-squared distribution equals 0.05. For 8 degrees of freedom the critical value equals 13.362.

Question 56

The 5^th percentile of a chi-squared distribution with 10 degrees of freedom is equal to 4.3903.

Question 57

The chi-squared critical value denotes the number on the measurement axis such that 10% of the area under the chi-squared curve with 6 degrees of freedom lies to the right of .

Question 58

The printing time of a weekly magazine was studied using two different machines, and . Eight different magazines were randomly assigned to each of the two printing machines, and produced standard deviations = 2.08, and = 1.66 for machines and , respectively. Assume the required assumptions are met and use = 0.05 to determine if the variances for the two machines are different.
Test Statistic = ______________
Reject Region: Reject H₀ if F > ______________
Conclusion: ______________
There is ______________ to indicate a difference in the population variances.

Question 59

If you wish to test vs. at the .05 level of significance using a sample of 20 observations, the critical values to be used are 32.852.

Question 60

The chi-square distribution can be used in constructing confidence intervals and carrying out hypothesis tests regarding the value of a population variance.

Question 61

In testing vs. the following sample data were recorded: 5.0, 6.1, and 11.1. Then, the value of the test statistic is:

A) 10.570
B) 5.285
C) 7.400
D) 6.714
E) none of these

Question 62

If a hypothesis test is to be conducted regarding a population variance, then the critical value of the test will be obtained using the:

A) normal distribution
B) t-distribution with n - 1 degrees of freedom
C) chi-square distribution with n - 1 degrees of freedom
D) binomial distribution
E) t-distribution with n + 1 degrees of freedom

Question 63

A pasta company would like to know the variability in the shelf life of their product. A random sample of 20 packages of pasta yielded a standard deviation of 5 days. Estimate the population variance using a 90% confidence interval. Assume the distribution is normal.
______________ Enter (n1, n2)

Question 64

A random sample of 20 observations is selected from a normally distributed population. The sample variance is 12. In the 95% confidence interval for the population variance, the upper limit will be:

A) 6.940
B) 7.564
C) 22.536
D) 25.599
E) 27.392

Question 65

A random sample of size 20 taken from a normally distributed population resulted in a sample variance of 32. The lower limit of a 90% confidence interval for the population variance would be:

A) 52.185
B) 20.375
C) 20.170
D) 54.931
E) 29.655

Question 66

If you are testing vs. using a sample of 15 observation and a significance level equal to .05, then the critical value of the test is:

A) 1.7613
B) 6.6450
C) 23.685
D) 4.867
E) 5.319

Question 67

A random sample of 25 observations is selected from a normally distributed population. The sample variance is 10. In the 95% confidence interval for the population variance, the upper limit will be:

A) 17.110
B) 6.097
C) 17.331
D) 19.353
E) 11.842

Question 68

In a hypothesis test for the population variance, the hypotheses are vs. . If the sample size is 15 and the test is being carried out at the 10% level of significance, the rejection region will be:

A) < 6.571 or> 23.685
B) < 7.790 or> 21.064
C) < 8.547 or> 22.307
D) < 7.261 or< 24.996
E) < 6.431 or> 21.653

Question 69

The sampling distribution used to make inferences about a single population variance is:

A) a normal distribution
B) a t distribution
C) an F distribution
D) a chi-square distribution
E) none of these

Question 70

The variability in a scientist's measuring equipment was observed for a random sample of 26 test runs from a normal distribution. The sample yielded a variance of 33. Estimate the population variance using a 95% confidence interval.
______________ Enter (n1, n2)

Question 71

The statistic is chi-squared distributed with n-1 degrees of freedom only if:

A) the population is normally distributed with variance equal to
B) the sample is normally distributed with variance equal to
C) the sample has a Student t distribution with degrees of freedom equal to n-1
D) all of these
E) none of these

Question 72

A car salesperson tries to convince customers buying a new Honda Accord that the gas mileage varies by only 2 miles per gallon when driving in town and on the highway. The customer found, for 14 randomly chosen tanks of gas, the gas mileage to vary by 2.8 miles per gallon. Do the customer's results contradict the salesperson's claim? Test the relevant hypotheses at the 0.05 level of significance. Assume the sample comes from a normal population.
Test Statistic = ______________
Reject Region: Reject H₀ if > ______________
Conclusion: ______________
Interpretation: ________________________________________________________

Question 73

A city bus driver claims that the route from the Canyon Crest Town Center to the University of California, Riverside never varies more than 1.3 minutes. A random sample of 18 trips yielded a variance of 1.26 minutes². It is of interest to determine if the sample data present sufficient evidence to reject the driver's claim. Calculate the value of the test statistic. What is the appropriate conclusion at the 0.05 level of significance?
Test Statistic = ______________
Reject Region: Reject H₀ if > ______________
Conclusion: ______________
One ______________ conclude that the route varies by more than 1.3 minutes.

Question 74

A random sample of size 25 taken from a normally distributed population resulted in a sample standard deviation of a 0.93054. The lower and upper limits of a 99% confidence interval for the population variance would be:

A) 9.886 and 45.559
B) 3.144 and 6.750
C) 0.678 and 1.449
D) 0.456 and 2.102
E) 1.493 and 6.430

Question 75

The shape of the chi-squared distribution depends on:

A) the population variance
B) the number of its degrees of freedom
C) the population mean
D) all of these
E) none of these

Question 76

The chi-squared distribution is:

A) symmetrical about 0
B) positively skewed ranging between 0 and
C) symmetrical around
D) negatively skewed ranging between -and 0
E) mound-shaped

Question 77

Like that of the Student t-distribution, the shape of the chi-squared distribution depends on:

A) the population size
B) the number of its degrees of freedom
C) the population standard deviation
D) whether the population is unimodal or bimodal
E) none of these

Question 78

Which of the following statements is true with regard to a hypothesis test for a population variance ?

A) The null hypothesis is stated in terms of.
B) The chi-square distribution is used to determine the critical value.
C) The critical value increases as the sample size increases for a given level of significance.
D) All of these.
E) None of these.

Question 79

The sampling distribution of the quantity is:

A) an F distribution
B) a chi-square distribution
C) a normal distribution
D) a t-distribution
E) none of these

Question 80

In testing vs. the following sample data were recorded: 11.5, 6.5, and 5.4. Then, the p-value of the test is:

A) between .05 and .10
B) between .10 and .20
C) less than .05
D) greater than .20
E) less than .10