Problem 50 - Basic Statistics for Business and Economics with Student

Question

The policy of the Suburban Transit Authority is to add a bus route if more than 55 percent of the potential commuters indicate they would use the particular route. A sample of 70 commuters revealed that 42 would use a proposed route from Bowman Park to the downtown area. Does the Bowman-to- downtown route meet the STA criterion? Use the .05 significance level.

Step-by-Step Solution

Answer

Yes, the Bowman-to-downtown route meets the STA criterion, as the sample shows more than 55% would use it and the test statistic exceeds the critical value.

Step 1: Define the Hypotheses

First, define the null and alternative hypotheses. The null hypothesis, denoted as \( H_0 \), states that the proportion of commuters who would use the new bus route is 55% or less, i.e., \( p \leq 0.55 \). The alternative hypothesis, denoted as \( H_a \), states that the proportion is greater than 55%, i.e., \( p > 0.55 \).

Step 2: Determine the Sample Proportion

Compute the sample proportion \( \hat{p} \) using the formula \( \hat{p} = \frac{x}{n} \), where \( x = 42 \) (the number of commuters who would use the route) and \( n = 70 \) (the total number of surveyed commuters). Thus, \( \hat{p} = \frac{42}{70} = 0.6 \).

Step 3: Calculate the Test Statistic

Use the formula for the test statistic in a one-sample proportion test: \( z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}} \), where \( p_0 = 0.55 \). Substitute to find \( z = \frac{0.6 - 0.55}{\sqrt{\frac{0.55(0.45)}{70}}} \). Calculate to obtain the z-value.

Step 4: Find the Critical Value

Determine the critical value for a one-tailed test at the 0.05 significance level. A common z-table or statistical software can be used to find that the critical z-value (for z > critical value in a right-tailed test) is approximately 1.645.

Step 5: Compare Test Statistic and Critical Value

Compare the computed z-statistic to the critical value from the z-table. If the z-statistic is greater than the critical z-value, reject the null hypothesis.

Step 6: Conclusion

Based on the comparison, if the z-value calculated is greater than 1.645, the null hypothesis is rejected, meaning that enough evidence exists to support the STA's criterion that more than 55% will use the route. Otherwise, it fails to meet the criterion.

Key Concepts

Significance Level

In hypothesis testing, the significance level, often denoted as \( \alpha \), is crucial as it helps determine the threshold for deciding if an observed effect is statistically significant. For this exercise, we are using a 0.05 significance level.

Here's what it means:

• If the probability of observing our data, assuming the null hypothesis is true, is less than 5%, we reject the null hypothesis.
• This level of significance is a balance between Type I errors (false positives) and Type II errors (false negatives).
• In more practical terms, at a 0.05 level, there's a 5% risk of concluding that there is an effect when there is actually none.

Choosing the right significance level is like setting a standard. A 0.05 threshold is commonly used in many fields, including public transit policies, as in this exercise.

Proportion Test

A proportion test evaluates hypotheses about a population proportion. It helps to determine if the observed proportion significantly differs from a known or assumed value.

When performing a proportion test, follow these general steps:

• First, identify the sample proportion \( \hat{p} \). In this exercise, it was calculated as \( \hat{p} = \frac{42}{70} = 0.6 \).
• Compare this sample proportion to the population proportion \( p_0 \), which is hypothesized (in this case, 0.55).

This test is crucial because it uses the information from a sample to make inferences about a larger population. It's like zooming out from a small picture to make conclusions about the bigger image. In this context, you determine if more than 55% of commuters would use the bus route based on the sample.

Null and Alternative Hypotheses

The null and alternative hypotheses are the foundation of hypothesis testing. They frame what you're looking to test. Each hypothesis acts as a possible answer to your question.

Null Hypothesis (\( H_0 \)):

• Assumes no effect or change. In our exercise: the proportion (\( p \)) of commuters willing to use the bus route is 55% or less, i.e., \( p \leq 0.55 \).

Alternative Hypothesis (\( H_a \)):

• Poses that there is an effect or a change. Here, it presumes that the proportion is greater than 55%, i.e., \( p > 0.55 \).

These hypotheses are mutually exclusive. If the evidence from data is strong enough, you will reject the null hypothesis and accept the alternative. This helps in decision-making, like determining whether to establish a new bus route based on commuter interest.

Test Statistic Calculation

Calculating the test statistic is a critical step in determining whether the sample data provides enough evidence to reject the null hypothesis.

In the case of a proportion test, the test statistic (\( z \)) formula is:

• \( z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}} \)

Where:

• \( \hat{p} \) is the sample proportion, calculated here as \( 0.6 \).
• \( p_0 \) is the hypothesized population proportion value (0.55).
• \( n \) is the sample size (70 in this exercise).

Plugging in the values gives you the test statistic. Compare this \( z \)-value to the critical \( z \)-value from a standard normal distribution to make your final hypothesis decision.

The test statistic acts like a measuring stick. It helps you see how far your sample proportion is from the hypothesized proportion, in units of standard error. This aids in deciding whether the observed data is typical under the null hypothesis or not.