Exam 10: Inference From Small Samples

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

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 .

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:

Automobile insurance appraisers examine cars that have been involved in accidental collisions to assess the cost of repairs. An insurance executive is concerned that different appraisers produce significantly different assessments. In an experiment 10 cars that have recently been involved in accidents were shown to two appraisers. Each assessed the estimated repair costs. These results are shown below. Can the executive conclude at the 5% significance level that the appraisers differ in their assessments? Use the Data Analysis software if you prefer. Test statistic = ______________ Critical Value(s) = ______________ Conclusion: ______________ Interpretation: __________________________________________

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:

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.

We can design a matched pairs experiment when the data collected are:

Two samples of sizes 25 and 20 are independently drawn from two normal populations, where the unknown population variances are assumed to be equal. The number of degrees of freedom of the equal-variances t-test statistic is 44.

The manufacturer of a particular battery pack for laptop computers claims its battery pack can function for 8 hours, on the average, before having to be recharged. A random sample of 16 battery packs was selected and tested. The mean functioning time before having to be recharged was 7.2 hours with a standard deviation of 1.9 hours. Assuming the distribution of functioning times is approximately normal, find a 95% confidence interval for the true average functioning time before needing to be recharged. Interpret the interval in part (a). Based on the interval in part (a), can the manufacturer's claim be rejected? Justify your answer. a. What is the 95% confidence interval (CI)? CI = ______________ Enter (n1, n2) Interpret the interval: ________________________________________________________ The claim ______________ be rejected. Justify your answer

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

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.

If a sample has 10 observations and a 90% confidence estimate for is needed, the appropriate t-score is 1.833.

The public relations officer for a particular city claims the average monthly cost for childcare outside the home for a single child is $600. A potential resident is interested in whether the claim is correct. She obtains a random sample of 14 records and computes the average monthly cost of this type of childcare to be $589 with a standard deviation of $40. Perform the appropriate test of hypothesis for the potential resident using . Approximate the p-value for the test in (a). a. Test Statistic = ______________ b. Compute the approximate p-value associated with the test statistic in (a). c. What is the p-value? ______________ Conclusion: ______________ The sample data ______________ support the null hypothesis at the level?

Student's t-distribution is a sampling distribution for a random variable, t, derived from a normally distributed population, that is (1) single-peaked above the random variable's mean, median, and mode of zero, (2) perfectly symmetrical about this central value, and (3) characterized by tails extending indefinitely in both directions from the center, approaching, but never touching, the horizontal axis.

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

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)

Given a random variable that has a t-distribution with the specified degrees of freedom, what percentage of the time will its value fall in the indicated region? 15 degrees of freedom, between -2.131 and 2.131 ______________ (Enter as a decimal percent or use the % sign.) 19 degrees of freedom, between -2.539 and 2.539 ______________ (Enter as a decimal percent or use the % sign.) 23 degrees of freedom, between -1.319 and 1.319 ______________ (Enter as a decimal percent or use the % sign.) 10 degrees of freedom, between -3.169 and 3.169 ______________ (Enter as a decimal percent or use the % sign.)

A political analyst in Michigan surveys a random sample of registered Democrats and compares the results with those obtained from a random sample of registered Republicans. This would be an example of an experimental design called a paired-difference or matched pairs design.

A random sample of 7 observations was drawn from a normal population. The following summations were computed: Test the hypotheses H₀: = 8 vs. H₁: > 8 at the 1% significance level. Test statistic = ______________ Critical Value(s) = ______________ Conclusion: ______________ Interpretation: ______________

Which of the following correctly describes degrees of freedom?