Problem 23 - Basic Statistics for Business and Economics with Student

Question

The National Safety Council reported that 52 percent of American turnpike drivers are men. A sample of 300 cars traveling southbound on the New Jersey Turnpike yesterday revealed that 170 were driven by men. At the .01 significance level, can we conclude that a larger proportion of men were driving on the New Jersey Turnpike than the national statistics indicate?

Step-by-Step Solution

Answer

No, there is not enough evidence to conclude a larger proportion of men were driving.

Step 1: Identify the Hypotheses

First, we need to set up the null and alternative hypotheses for the problem. The null hypothesis (H0) is that the proportion of men driving is equal to the national average, which is 0.52. The alternative hypothesis (H1) is that the proportion of men driving is greater than 0.52. Mathematically, these are notated as: \[ H_0: p = 0.52 \]\[ H_1: p > 0.52 \]

Step 2: Determine Significance Level

We are given a significance level \( \alpha = 0.01 \). This is the threshold for deciding whether to reject the null hypothesis.

Step 3: Calculate Sample Proportion

The sample proportion \( \hat{p} \) is calculated as the number of men drivers divided by the total number of drivers observed: \[ \hat{p} = \frac{170}{300} = 0.5667 \]

Step 4: Calculate Standard Error

The standard error (SE) of the proportion can be calculated using the formula: \[ SE = \sqrt{\frac{p(1-p)}{n}} \]where \( p = 0.52 \) and \( n = 300 \). Therefore, \[ SE = \sqrt{\frac{0.52 \times 0.48}{300}} \approx 0.0288 \]

Step 5: Compute the Test Statistic

The test statistic (Z) is calculated using the formula: \[ Z = \frac{\hat{p} - p}{SE} \]Substituting the known values, we have:\[ Z = \frac{0.5667 - 0.52}{0.0288} \approx 1.6215 \]

Step 6: Determine Critical Value

For a one-tailed test at \( \alpha = 0.01 \), the critical value from the Z-distribution is approximately 2.33.

Step 7: Make the Decision

Since the calculated Z value of 1.6215 is less than the critical Z value of 2.33, we do not reject the null hypothesis. This means there is not enough evidence to conclude that a larger proportion of men were driving.

Key Concepts

Proportion Test

A proportion test is a statistical method used to decide if the proportion of a specific characteristic in a sample significantly differs from a known or assumed population proportion. In the context of the provided problem, we aim to determine if the proportion of men driving on the New Jersey Turnpike is greater than the national proportion of 0.52. This process involves setting up hypotheses, calculating a sample proportion, and making statistical comparisons.

• **Null Hypothesis (H0):** The proportion of men driving equals the national average (0.52).
• **Alternative Hypothesis (H1):** The proportion of men driving is greater than 0.52, showing a difference.

The goal of a proportion test is to analyze sample data to see if there is statistical evidence strong enough to support the alternative hypothesis.

Significance Level

The significance level, denoted by \( \alpha \), is a crucial threshold in hypothesis testing that defines the probability of rejecting the null hypothesis when in fact, it is true. A common standard in many scenarios uses a significance level such as 0.01, 0.05, or 0.10. In our exercise, the significance level is set at 0.01.Choosing a significance value of 0.01 indicates a stringent test, allowing only a 1% chance of incorrectly rejecting the null hypothesis, enhancing the reliability of results. When conducting a test:

• If the test result is sufficiently extreme compared to this set threshold, we may reject the null hypothesis.
• If not, we do not reject the null hypothesis, suggesting insufficient evidence to support the alternative hypothesis.

Standard Error

Standard Error (SE) is an essential measure in statistics that quantifies the amount of variation or "noise" around a parameter estimate such as a mean or proportion. In this context, the SE is used to assess the variability of the sample proportion from the true proportion of the population.The formula for calculating the Standard Error of a proportion is given by:\[SE = \sqrt{\frac{p(1-p)}{n}}\]where \( p \) is the population proportion, and \( n \) is the sample size. In our example, with \( p = 0.52 \) and \( n = 300 \), the calculated SE is approximately 0.0288.A smaller SE indicates that the sample mean is an accurate reflection of the actual population mean. It provides a basis to assess how much the sample statistic might "jitter" around the true population parameter.

Test Statistic

The test statistic is a standardized value that helps in determining the direction and extent of the deviation of the sample statistic from the null hypothesis. In proportion tests, this is often represented in the form of a Z-score, calculated by comparing the sample proportion to the population proportion, scaled by the standard error:\[Z = \frac{\hat{p} - p}{SE}\]where \( \hat{p} \) is the sample proportion, \( p \) is the population proportion, and \( SE \) is the standard error.In our exercise:

• The sample proportion \( \hat{p} \) is 0.5667, derived from observing 170 male drivers out of 300.
• The Z-score computed here is approximately 1.6215.

The Z-score helps us determine how many standard deviations the sample proportion is away from the hypothesized population proportion. It is compared against a critical value from the Z-distribution to conclude if there is enough statistical evidence to reject the null hypothesis.