Question

Perform these steps. a. State the hypotheses and identify the claim. b. Find the critical value(s). c. Compute the test value. d. Make the decision. e. Summarize the results. Use the traditional method of hypothesis testing unless otherwise specified. National statistics show that \(23 \%\) of men smoke and \(18.5 \%\) of women smoke. A random sample of 180 men indicated that 50 were smokers, and a random sample of 150 women surveyed indicated that 39 smoked. Construct a \(98 \%\) confidence interval for the true difference in proportions of male and female smokers. Comment on your interval-does it support the claim that there is a difference?

Short answer

There is not enough evidence to conclude a significant difference in smoking rates between men and women.

Step-by-step solution

State the Hypotheses and Identify the Claim

We want to determine if there is a difference in the proportions of male and female smokers. Therefore, we define our null hypothesis as:\[ H_0: p_1 - p_2 = 0 \]Where \( p_1 \) is the proportion of men who smoke, and \( p_2 \) is the proportion of women who smoke.The alternative hypothesis is:\[ H_a: p_1 - p_2 eq 0 \]The claim we are testing is that there is a difference in the proportions, which corresponds to the alternative hypothesis.

Find the Critical Value(s)

For a two-tailed test with \( \alpha = 0.02 \) (since 98% confidence level means 2% significance level), we need to find the critical z-value. Using a standard normal distribution table or calculator, the critical values for a 98% confidence interval are approximately \( \pm 2.33 \).

Compute the Test Value

First, calculate the sample proportions:\[ \hat{p}_1 = \frac{50}{180} \approx 0.2778 \]\[ \hat{p}_2 = \frac{39}{150} \approx 0.26 \]Next, compute the pooled sample proportion:\[ \hat{p} = \frac{50+39}{180+150} = \frac{89}{330} \approx 0.2697 \]Now calculate the standard error (SE):\[ SE = \sqrt{\hat{p}(1 - \hat{p}) \left( \frac{1}{n_1} + \frac{1}{n_2} \right)} = \sqrt{0.2697 \times 0.7303 \left( \frac{1}{180} + \frac{1}{150} \right)} \]\[ SE \approx 0.0582 \]Compute the test statistic (z):\[ z = \frac{\hat{p}_1 - \hat{p}_2}{SE} = \frac{0.2778 - 0.26}{0.0582} \approx 0.307 \]

Make the Decision

Compare the computed test statistic to the critical value(s). The calculated z-value is 0.307. Since 0.307 is within the range of -2.33 and 2.33, we fail to reject the null hypothesis \( H_0 \).

Summarize the Results

The 98% confidence interval calculation would confirm that there is not enough evidence at the .02 level of significance to reject the null hypothesis. Therefore, the claim that there is a significant difference between the proportions of male and female smokers is not supported by the data.

Key concepts

Confidence Interval

A confidence interval is a statistical tool used to estimate the range within which a population parameter lies, based on sample data. In this context, it helps us understand the range of possible differences in smoking rates between men and women. The confidence interval is constructed by using the sample proportions and standard error. We are given a 98% confidence level, which implies a 2% significance level (or \(\alpha = 0.02\)). This high level of confidence indicates that we are 98% certain that the true difference in proportions falls within our calculated range.To calculate the confidence interval for the difference in proportions, we need to determine the sample proportions for both groups and then utilize these to find the interval:- Calculate the sample proportion for each group (men and women).- Use the formula for the standard error of the difference in proportions.- Apply the critical value for the given confidence level to find the confidence interval.Ultimately, the confidence interval will tell us whether the true difference is likely to include zero, which implies no significant difference if it does.

Difference in Proportions

The difference in proportions is a way to measure how two groups compare in terms of a certain characteristic—in this case, smoking rates. This is a useful statistic when you want to evaluate whether one group is statistically different from another.To compute the difference in sample proportions:- Calculate the sample proportion of smokers in each group: \( \hat{p}_1 \) for men and \( \hat{p}_2 \) for women.- The difference in sample proportions is simply \( \hat{p}_1 - \hat{p}_2 \).In this exercise, we find the sample proportion of male smokers to be approximately \(0.2778\) and that of female smokers to be approximately \(0.26\). The difference is used to explore whether this observed discrepancy is statistically meaningful or could have occurred by random chance alone. A hypothesis test allows us to formally assess this possibility.

Critical Value

In hypothesis testing, a critical value helps determine the threshold at which we either reject or fail to reject the null hypothesis. It corresponds to the test's confidence level; here, a 98% confidence level translates to a critical z-value.For a two-tailed test with \(\alpha = 0.02\), we find the z-values that leave 1% in each tail of the normal distribution. From statistical tables or using a calculator, this results in a critical value of approximately \( \pm 2.33 \).Critical values are essential as they define the boundary within which we decide the fate of our hypothesis. If the test statistic falls beyond these bounds, it suggests that the observed data is incompatible with the null hypothesis, prompting its rejection.

Test Statistic

The test statistic is a standardized value that helps compare the observed data against what we would expect under the null hypothesis. It combines sample data with theoretical properties to determine whether to reject the null hypothesis.In the case of comparing proportions, the test statistic (z) is calculated as follows:- Determine the sample proportions \(\hat{p}_1\) and \(\hat{p}_2\).- Calculate the pooled sample proportion \(\hat{p}\) for combined groups.- Find the standard error based on these proportions.- Compute the z-value: \[ z = \frac{\hat{p}_1 - \hat{p}_2}{SE} \]Using the given data, the computed z-value is approximately 0.307. This value is then compared to the critical values from the hypothesis test. Since 0.307 is between -2.33 and 2.33, we fail to reject the null hypothesis, indicating insufficient evidence for a significant difference in smoking rates between men and women.