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Chapter 13: Problem 5

The paper "Ladies First?" A Field Study of Discrimination in Coffee Shops" (Applied Economics [2008]: 1-19) describes a study in which researchers observed wait times in coffee shops in Boston. Both wait time and gender of the customer were observed. The mean wait time for a sample of 145 male customers was 85.2 seconds. The mean wait time for a sample of 141 female customers was 113.7 seconds. The sample standard deviations (estimated from graphs in the paper) were 50 seconds for the sample of males and 75 seconds for the sample of females. Suppose that these two samples are representative of the populations of wait times for female coffee shop customers and for male coffee shop customers. Is there convincing evidence that the mean wait time differs for males and females? Test the relevant hypotheses using a significance level of 0.05

Short Answer

Expert verified
The solution involves calculating the z-test statistic and comparing it with the critical z-value to see if the null hypothesis can be rejected, thus providing evidence of a difference in mean wait times between males and females.

Step by step solution

01

Formulate the null hypothesis and the alternate hypothesis

The null hypothesis (H0) is that the mean wait time of males(M1) and females(M2) is the same: \( H0: M1 = M2 \).\nThe alternate hypothesis (HA) is that the mean wait times differ: \( HA: M1 \neq M2 \).
02

Calculate the test statistic and the critical z-score

The test statistic z can be calculated using the formula: \[ Z = \frac{{(M1 - M2) - D0}}{{\sqrt{{\frac{{S1^2}}{{n1}} + \frac{{S2^2}}{{n2}}}}}} \] \nWhere M1 and M2 are the sample means, D0 is the hypothesized difference, S1 and S2 are the standard deviations, and n1 and n2 are the number of observations. Here, M1 is 85.2, M2 is 113.7, D0 is 0, S1 = 50, n1 = 145, S2 = 75, n2 =141. The critical z value at a significance level of 0.05 for a two-tailed test is \(\pm 1.96\).
03

Conduct the hypothesis test

If the calculated z-score falls within the critical region (-1.96 to 1.96), reject the null hypothesis. Otherwise, we do not have enough evidence to reject the null hypothesis.
04

Interpret the Result

If the null hypothesis is rejected, there is significant evidence to suggest the mean wait time differs for males and females. If not, there is insufficient evidence to suggest the mean wait times differ.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Null Hypothesis
In hypothesis testing, the null hypothesis (denoted as \( H_0 \)) is a statement that there is no effect or no difference in a population. It's the starting point for statistical testing.

For example, in the coffee shop study, the null hypothesis states that the mean wait time for male customers \( M_1 \) is equal to the mean wait time for female customers \( M_2 \). In other words:
  • \( H_0: M_1 = M_2 \)
By assuming that there's no difference (\( D_0 = 0 \)), the null hypothesis serves as a baseline to compare the data against. It's only rejected when we have enough evidence to support an alternative explanation.
Alternative Hypothesis
The alternative hypothesis (denoted as \( H_A \)) is a statement that contradicts the null hypothesis. It suggests that there is an effect or a difference in the population.

In the context of the coffee shop study, the alternative hypothesis posits that the mean wait times for male and female customers are not the same:
  • \( H_A: M_1 eq M_2 \)
This hypothesis is considered if there's sufficient statistical evidence to reject the null hypothesis. Essentially, it represents the finding that the researcher is trying to prove.
Test Statistic
A test statistic is a standardized value that helps us decide whether to reject the null hypothesis. It measures how far the sample statistic is from the hypothesis in terms of standard errors.

To calculate the test statistic in a situation like this, we use the formula: \[ Z = \frac{{(M_1 - M_2) - D_0}}{{\sqrt{{\frac{{S_1^2}}{{n_1}} + \frac{{S_2^2}}{{n_2}}}}}} \]
  • \( M_1 \) and \( M_2 \) are the sample means of the two groups.
  • \( S_1 \) and \( S_2 \) are the standard deviations.
  • \( n_1 \) and \( n_2 \) are the sample sizes.
  • \( D_0 \) is the hypothesized difference (usually zero).
This results in a z-score, which we then compare to critical values to make a decision about the hypothesis.
Significance Level
The significance level, often denoted by \( \alpha \), is a threshold set to determine when to reject the null hypothesis. It's the probability of incorrectly rejecting the null hypothesis when it's true (a Type I error).

Commonly set at 0.05, it means that there's a 5% chance of making a wrong decision.

In the coffee shop study, the significance level is 0.05, corresponding to critical z values of \( \pm 1.96 \) for a two-tailed test. If the calculated z-score falls outside this range, the null hypothesis is rejected, suggesting a significant difference in waiting times between genders. This balance between risking an error and gaining insights is what makes hypothesis testing powerful.

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Most popular questions from this chapter

Some people believe that talking on a cell phone while driving slows reaction time, increasing the risk of accidents. The study described in the paper "A Comparison of the Cell Phone Driver and the Drunk Driver" (Human Factors [2006]: \(381-391\) ) investigated the braking reaction time of people driving in a driving simulator. Drivers followed a pace car in the simulator, and when the pace car's brake lights came on, the drivers were supposed to step on the brake. The time between the pace car brake lights coming on and the driver stepping on the brake was measured. Two samples of 40 drivers participated in the study. The 40 people in one sample used a cell phone while driving. The 40 people in the second sample drank a mixture of orange juice and alcohol in an amount calculated to achieve a blood alcohol level of \(0.08 \%\) (a value considered legally drunk in most states). For the cell phone sample, the mean braking reaction time was 779 milliseconds and the standard deviation was 209 milliseconds. For the alcohol sample, the mean breaking reaction time was 849 milliseconds and the standard deviation was \(228 .\) Is there convincing evidence that the mean braking reaction time is different for the population of drivers talking on a cell phone and the population of drivers who have a blood alcohol level of \(0.08 \%\) ? For purposes of this exercise, you can assume that the two samples are representative of the two populations of interest.

An individual can take either a scenic route to work or a nonscenic route. She decides that use of the nonscenic route can be justified only if it reduces the mean travel time by more than 10 minutes. a. If \(\mu_{1}\) refers to the mean travel time for scenic route and \(\mu_{2}\) to the mean travel time for nonscenic route, what hypotheses should be tested? b. If \(\mu_{1}\) refers to the mean travel time for nonscenic route and \(\mu_{2}\) to the mean travel time for scenic route, what hypotheses should be tested?

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Dentists make many people nervous. To see whether such nervousness elevates blood pressure, the blood pressure and pulse rates of 60 subjects were measured in a dental setting and in a medical setting ( \({ }^{4}\) The Effect of the Dental Setting on Blood Pressure Measurement," American Journal of Public Health [1983]: \(1210-1214\) ). For each subject, the difference (dental setting blood pressure minus medical setting blood pressure) was calculated. The (dental - medical) differences were also calculated for pulse rates. Summary statistics follow. $$ \begin{array}{lcc} & & \text { Standard } \\ & \begin{array}{c} \text { Mean } \\ \text { Difference } \end{array} & \begin{array}{c} \text { Deviation of } \\ \text { Differences } \end{array} \\ \text { Systolic Blood Pressure } & 4.47 & 8.77 \\ \text { Pulse (beats/min) } & -1.33 & 8.84 \end{array} $$

Do children diagnosed with attention deficit/ hyperactivity disorder (ADHD) have smaller brains than children without this condition? This question was the topic of a research study described in the paper "Developmental Trajectories of Brain Volume Abnormalities in Children and Adolescents with Attention Deficit/Hyperactivity Disorder" (journal of the American Medical Association [2002]: \(1740-\) 1747). Brain scans were completed for a representative sample of 152 children with ADHD and a representative sample of 139 children without ADHD. Summary values for total cerebral volume (in milliliters) are given in the following table: $$ \begin{array}{lccc} & n & \bar{x} & s \\ \hline \text { Children with ADHD } & 152 & 1,059.4 & 117.5 \\ \text { Children Without ADHD } & 139 & 1,104.5 & 111.3 \end{array} $$ Use a \(95 \%\) confidence interval to estimate the differ- ence in mean brain volume for children with and without ADHD.
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