9. Hypothesis Testing for One Sample

Bootstrapping and Randomization When resampling data from two independent samples, what is the fundamental difference between bootstrapping and randomization?

Notation Using the weights (lb) and highway fuel consumption amounts (mi/gal) of the 48 cars listed in Data Set 35 “Car Data” of Appendix B, we get this regression equation:

y^ = 58.9 - 0.00749x, where x represents weight.

a. What does the symbol y^ represent?

Notation Using the weights (lb) and highway fuel consumption amounts (mi/gal) of the 48 cars listed in Data Set 35 “Car Data” of Appendix B, we get this regression equation:

y^ = 58.9 - 0.00749x, where x represents weight.

c. What is the predictor variable?

Notation What is the difference between the regression equation y^ = b0 + b1x and the regression equation y = β0 + β1x.

Best-Fit Line

What is a residual?

In what sense is the regression line the straight line that “best” fits the points in a scatterplot?

One-Way ANOVA In general, what is one-way analysis of variance used for?

Interaction

a. What is an interaction between two factors?

Interaction

b. In general, when using two-way analysis of variance, if we find that there is an interaction effect, how does that affect the procedure?

Sampling Methods A student obtains a sample of responses to the question “Do you plan to take or have you taken a statistics course?” A second student obtains a sample of responses to the same question. The first student surveys only males at the same college, and the second student surveys only females at the same college. What is wrong with the samples? Can randomization be used to overcome the flaws of those samples?

Testing Claims About Variation

In Exercises 5–16, test the given claim. Identify the null hypothesis, alternative hypothesis, test statistic, P-value, or critical value(s), then state the conclusion about the null hypothesis, as well as the final conclusion that addresses the original claim. Assume that a simple random sample is selected from a normally distributed population.

Bank Lines The Jefferson Valley Bank once had a separate customer waiting line at each teller window, but it now has a single waiting line that feeds the teller windows as vacancies occur. The standard deviation of customer waiting times with the old multiple-line configuration was 1.8 min. Listed below is a simple random sample of waiting times (minutes) with the single waiting line. Use a 0.05 significance level to test the claim that with a single waiting line, the waiting times have a standard deviation less than 1.8 min. What improvement occurred when banks changed from multiple waiting lines to a single waiting line?

6.5 6.6 6.7 6.8 7.1 7.3 7.4 7.7 7.7 7.7

RESAMPLING

a. In general, what does it mean to “resample” the following data set consisting of wait times (minutes) of customers waiting in line for the Space Mountain ride at Walt Disney World: 50, 25, 75, 35, 50?

Finding P-values

In Exercises 5–8, either use technology to find the P-value or use Table A-3 to find a range of values for the P-value. Based on the result, what is the final conclusion?

Cotinine in Smokers The claim is that smokers have a mean cotinine level greater than the level of 2.84 ng/mL found for nonsmokers. (Cotinine is used as a biomarker for exposure to nicotine.) The sample size is n = 902 and the test statistic is t = 56.319.

Testing Claims About Variation

In Exercises 5–16, test the given claim. Identify the null hypothesis, alternative hypothesis, test statistic, P-value, or critical value(s), then state the conclusion about the null hypothesis, as well as the final conclusion that addresses the original claim. Assume that a simple random sample is selected from a normally distributed population.

Birth Weights A simple random sample of birth weights of 30 girls has a standard deviation of 829.5 g. Use a 0.01 significance level to test the claim that birth weights of girls have the same standard deviation as birth weights of boys, which is 660.2 g (based on Data Set 6 “Births” in Appendix B).

Minting Dollar Coins For the sample data from Exercise 1, we get a P-value of 0.0041 when testing the claim that σ < 0.04000 g.

What should we conclude about the null hypothesis?

What should we conclude about the original claim?

What do these results suggest about the new minting process?

Finding Critical Values of (chi)^2 For large numbers of degrees of freedom, we can approximate critical values of as follows:

(chi)^2 = (1/2)(z + sqrt(2k-1))

Here k is the number of degrees of freedom and z is the critical value(s) found from technology or Table A-2. In Exercise 12 “Spoken Words” we have df = 55, so Table A-4 does not list an exact critical value. If we want to approximate a critical value of (chi)^2 in the right-tailed hypothesis test with α = 0.01 and a sample size of 56, we let k =55 with z = 2.33 (or the more accurate value of z = 2.326348 found from technology). Use this approximation to estimate the critical value of for Exercise 12. How close is it to the critical value of (chi)^2 = 82.292 obtained by using Statdisk and Minitab?