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Chapter 23: Problem 27

The survey in Exercise 23.26 also looked at possible differences in the proportions of males and females who used ride-hailing apps. They found that 378 of the 2361 males and 340 of the 2426 females had used a ride-hailing app. Is there evidence of a difference between the proportions of males and females who have used a ridehailing app?

Short Answer

Expert verified
No significant difference in proportions of males and females using ride-hailing apps.

Step by step solution

01

Define the Hypotheses

Let's define the null hypothesis \((H_0)\) and the alternative hypothesis \((H_a)\). The null hypothesis is that there is no difference between the proportions of males \((p_m)\) and females \((p_f)\) who have used a ride-hailing app: \(H_0: p_m = p_f\). The alternative hypothesis is that there is a difference: \(H_a: p_m eq p_f\).
02

Calculate Sample Proportions

Calculate the sample proportion of males who used the app: \( \hat{p}_m = \frac{378}{2361} \approx 0.160\). Calculate the sample proportion of females who used the app: \( \hat{p}_f = \frac{340}{2426} \approx 0.140\).
03

Find the Combined Proportion

Calculate the combined proportion \(\hat{p}\) of users: \( \hat{p} = \frac{378 + 340}{2361 + 2426} = \frac{718}{4787} \approx 0.150\). This is the overall proportion of survey participants who used the app.
04

Calculate the Standard Error

Use the combined proportion \(\hat{p}\) to calculate the standard error \(SE\): \[ SE = \sqrt{\hat{p}(1-\hat{p})\left(\frac{1}{n_1} + \frac{1}{n_2}\right)} \] Plug in the numbers: \(SE = \sqrt{0.150(1-0.150)\left(\frac{1}{2361} + \frac{1}{2426}\right)} \approx 0.011\).
05

Compute the Test Statistic

Calculate the test statistic \(z\) using the formula: \[ z = \frac{\hat{p}_m - \hat{p}_f}{SE} \] Plug in the sample proportions and standard error: \(z = \frac{0.160 - 0.140}{0.011} \approx 1.818\).
06

Determine the Critical Value

For a two-tailed test at a standard significance level of 5% (\(\alpha = 0.05\)), the critical z-value is approximately 1.96. This comes from a standard normal distribution table
07

Make a Decision

The calculated test statistic \(z = 1.818\) does not exceed the critical value of \(1.96\). Therefore, we do not reject the null hypothesis.

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Key Concepts

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

Sample Proportions
In hypothesis testing, particularly when examining proportions, understanding sample proportions is crucial. A sample proportion (\( \hat{p} \) ) is a statistical measure that represents the fraction or percentage of a subset of a population exhibiting a particular trait or behavior.
For example, in our study, we found sample proportions to determine how many males and females used ride-hailing apps.
To get these proportions:
  • Divide the number of individuals with the trait by the total number of individuals in the sample.
  • For males: 378 used the app out of 2361 sampled, thus \( \hat{p}_m = \frac{378}{2361} \approx 0.160 \).
  • For females: 340 used the app out of 2426, thus \( \hat{p}_f = \frac{340}{2426} \approx 0.140 \).
Sample proportions are foundational in analyzing whether differences in behavior or characteristics in distinct groups within a population are significant or merely due to random chance.
Standard Error
The standard error (SE) measures the variability or spread of a sampling distribution. It's essentially the standard deviation of sample proportions.
When comparing two proportions, the standard error helps us understand the variation we'd expect if we repeatedly drew samples from the same populations.
To calculate the SE when comparing two groups:
  • First, compute the combined proportion (\( \hat{p} \)) of the event occurring in both groups combined.
  • Using \( \hat{p} = \frac{718}{4787} \approx 0.150 \) (from combining our male and female samples), we then apply the SE formula:\[SE = \sqrt{\hat{p}(1-\hat{p})\left(\frac{1}{n_1} + \frac{1}{n_2}\right)}\]
  • For our case, \( SE \approx 0.011 \).
The smaller the SE, the more consistent the sample proportions are, indicating higher precision in our estimation of the population proportion difference.
Test Statistic
A test statistic is a standardized value that helps in deciding whether to reject the null hypothesis. It measures how far our observed sample statistic is from the null hypothesis' expectation.
In this context, the test statistic is calculated by considering the difference between sample proportions, divided by the standard error.
The formula used here is:\[z = \frac{\hat{p}_m - \hat{p}_f}{SE}\]
  • Using the previously calculated proportions (\( \hat{p}_m \) and \( \hat{p}_f \)) and SE:
  • \( z = \frac{0.160 - 0.140}{0.011} \approx 1.818 \).
This z-score tells us how many standard deviations away our observed difference is from the assumed population difference of zero under the null hypothesis. The test statistic is then compared against a critical value to decide on rejecting or failing to reject the null hypothesis.
Null and Alternative Hypotheses
Hypothesis testing starts with defining the null and alternative hypotheses. These are statements about a population parameter we aim to test.
  • The **null hypothesis** (\( H_0 \)) suggests no effect or no difference. In our case: \( H_0: p_m = p_f \).
  • The **alternative hypothesis** (\( H_a \)) is what we seek evidence for; a difference exists: \( H_a: p_m eq p_f \).
The goal of hypothesis testing is to see if there is sufficient statistical evidence to support the alternative hypothesis over the null hypothesis. It's important to note that failing to reject \( H_0 \) does not prove \( H_0 \) true; it merely suggests inadequate evidence to support \( H_a \).In our study, since the test statistic did not exceed the critical value, we fail to reject \( H_0 \), meaning no significant difference was detected between the proportions of males and females using ride-hailing apps.

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