BackGender Wage Gap It has long been a concern that there is a wage gap between men and women in the United States with some reports suggesting that women only make \$0.77 for every dollar earned by a man. Design a study that would allow you to confirm whether a wage gap does actually exist.
In Problem 3, assume that the paired differences come from a population that is normally distributed.
a. Compute di=Xi−Yi for each pair of data.
McDonald’s versus Wendy’s A student wanted to determine whether the wait time in the drive-thru at McDonald’s differed from that at Wendy’s. She used a random sample of 30 cars at McDonald’s and 27 cars at Wendy’s and obtained these results:
c. Is there a difference in wait times at each restaurant’s drive-thru? Use the α=0.1 level of significance.
Note: The sample size for Wendy’s is less than 30. However, the data do not contain any outliers, so the Central Limit Theorem can be used.
a. Determine dᵢ = Xᵢ - Yᵢ for each pair of data.
Naughty or Nice? An experiment was conducted in which 16 ten-month-old babies were asked to watch a climber character attempt to ascend a hill. On two occasions, the baby witnesses the character fail to make the climb. On the third attempt, the baby witnesses either a helper toy push the character up the hill, or a hinderer toy preventing the character from making the ascent. The helper and hinderer toys were shown to each baby in a random fashion for a fixed amount of time. In Problem 41 from Section 10.2, we learned that, after watching both the helper and hinderer toy in action, 14 of 16 ten-month-old babies preferred to play with the helper toy when given a choice as to which toy to play with. A second part of this experiment showed the climber approach the helper toy, which is not a surprising action, and then alternatively the climber approached the hinderer toy, which is a surprising action. The amount of time the ten-month-old watched the event was recorded. The mean difference in time spent watching the climber approach the hinderer toy versus watching the climber approach the helper toy was 1.14 seconds with a standard deviation of 1.75 second. Source: J. Kiley Hamlin et al., “Social Evaluation by Preverbal Infants,” Nature, Nov. 2007.
c. What do you think the results of this experiment imply about 10-month-olds’ ability to assess surprising behavior?
[NW] [DATA] Muzzle Velocity The following data represent the muzzle velocity (in feet per second) of rounds fired from a 155-mm gun. For each round, two measurements of the velocity were recorded using two different measuring devices, with the following data obtained:
a. Why are these matched-pairs data?
Reaction Time In an experiment conducted online at the University of Mississippi, study participants are asked to react to a stimulus. In one experiment, the participant must press a key on seeing a blue screen and reaction time (in seconds) to press the key is measured. The same person is then asked to press a key on seeing a red screen, again with reaction time measured. The results for six randomly sampled study participants are as follows:
c. Is the reaction time to the blue stimulus different from the reaction time to the red stimulus at the α=0.01 level of significance? Note: A normal probability plot and boxplot of the data indicate that the differences are approximately normally distributed with no outliers.
[DATA] Invest in Education Go to www.pearsonhighered.com/sullivanstats to obtain the data file 12_3_17. The variable “Cost” represents the four-year cost including tuition, supplies, room and board, the variable “Annual ROI” represents the return on investment for graduates of the school—essentially how much you would earn on the investment of attending the school. The variable “Grad Rate” represents the graduation rate of the school.
a. In Problem 49 from Section 4.1, a scatter diagram between “Cost” and “Grad Rate” treating “Cost” as the explanatory variable suggested a positive association between the two variables. Treating “Cost” as the explanatory variable, x, test whether a negative association exists between the cost and annual ROI for graduates of four-year schools at the alpha = 0.01 level of significance. Normal probability plots suggest the residuals are normally distributed.
[DATA] Does Octane Matter? Octane is a measure of how much fuel can be compressed before it spontaneously ignites. Some people believe that higher-octane fuels result in better gas mileage for their cars. To test this claim, a researcher randomly selected 11 individuals (and their cars) to participate in the study. Each participant received 10 gallons of gas and drove his or her car on a closed course that simulated both city and highway driving. The number of miles driven until the car ran out of gas was recorded. A coin flip was used to determine whether the car was filled up with 87-octane or 92-octane fuel first, and the driver did not know which type of fuel was in the tank. The results are in the following table:
a. Why is it important that the matching be done by driver and car?
In Exercises 5–16, use the listed paired sample data, and assume that the samples are simple random samples and that the differences have a distribution that is approximately normal.
Measured and Reported Weights Listed below are measured and reported weights (lb) of random female subjects (from Data Set 4 “Measured and Reported” in Appendix B).
b. Construct the confidence interval that could be used for the hypothesis test described in part (a). What feature of the confidence interval leads to the same conclusion reached in part (a)?
In Exercises 5–16, use the listed paired sample data, and assume that the samples are simple random samples and that the differences have a distribution that is approximately normal.
Do Men Talk Less than Women? Listed below are word counts of males and females in couple relationships (from Data Set 14 “Word Counts” in Appendix B).
b. Construct the confidence interval that could be used for the hypothesis test described in part (a). What feature of the confidence interval leads to the same conclusion reached in part (a)?
In Exercises 5–16, use the listed paired sample data, and assume that the samples are simple random samples and that the differences have a distribution that is approximately normal.
The Freshman 15 The “Freshman 15” refers to the belief that college students gain 15 lb (or 6.8 kg) during their freshman year. Listed below are weights (kg) of randomly selected male college freshmen (from Data Set 13 “Freshman 15” in Appendix B). The weights were measured in September and later in April.
a. Use a 0.01 significance level to test the claim that for the population of freshman male college students, the weights in September are less than the weights in the following April.
In Exercises 5–16, use the listed paired sample data, and assume that the samples are simple random samples and that the differences have a distribution that is approximately normal.
The Freshman 15 The “Freshman 15” refers to the belief that college students gain 15 lb (or 6.8 kg) during their freshman year. Listed below are weights (kg) of randomly selected male college freshmen (from Data Set 13 “Freshman 15” in Appendix B). The weights were measured in September and later in April.
c. What do you conclude about the Freshman 15 belief?
In Exercises 5–16, use the listed paired sample data, and assume that the samples are simple random samples and that the differences have a distribution that is approximately normal.
Heights of Presidents A popular theory is that presidential candidates have an advantage if they are taller than their main opponents. Listed are heights (cm) of presidents along with the heights of their main opponents (from Data Set 22 “Presidents” in Appendix B).
a. Use the sample data with a 0.05 significance level to test the claim that for the population of heights of presidents and their main opponents, the differences have a mean greater than 0 cm.
In Exercises 1–10, based on the nature of the given data, do the following:
a. Pose a key question that is relevant to the given data.
b. Identify a procedure or tool from this chapter or the preceding chapters to address the key question from part (a).
c. Analyze the data and state a conclusion.
IQ Scores of Twins Listed below are IQ scores of twins listed in Data Set 12 “IQ and Brain Size” in Appendix B. The data are pairs of IQ scores from ten different families.