91Ó°ÊÓ

Never forget that even small effects can be statistically significant if the samples are large. To illustrate this fact, consider a sample of 148 small businesses. During a three-year period, 15 of the 106 headed by men and 7 of the 42 headed by women failed. 22 (a) Find the proportions of failures for businesses headed by women and businesses headed by men. These sample proportions are quite close to each other. Give the \(P\)-value for the \(z\) test of the hypothesis that the same proportion of women's and men's businesses fail. (Use the two-sided alternative.) The test is very far from being significant. (b) Now suppose that the same sample proportions came from a sample 30 times as large. That is, 210 out of 1260 businesses headed by women and 450 out of 3180 businesses headed by men fail. Verify that the proportions of failures are exactly the same as in part (a). Repeat the \(z\) test for the new data, and show that it is now significant at the \(\alpha=0.05\) level. (c) It is wise to use a confidence interval to estimate the size of an effect rather than just giving a \(P\)-value. Give the large sample \(95 \%\) confidence intervals for the difference between the proportions of women's and men's businesses that fail for the settings of both parts (a) and (b). What is the effect of larger samples on the confidence interval?

Step by step solution

01

Calculate Proportions of Failures

For businesses headed by women, the proportion of failures is \( \frac{7}{42} = 0.1667 \). For businesses headed by men, the proportion of failures is \( \frac{15}{106} = 0.1415 \).

02

Perform the z Test (Part a)

The null hypothesis \( H_0 \) is that the proportions are equal, \( p_1 = p_2 \). Using a two-proportion z test, we calculate the pooled proportion \( \hat{p} = \frac{7 + 15}{42 + 106} = 0.1493 \). The z test statistic is \( z = \frac{0.1667 - 0.1415}{\sqrt{0.1493 (1 - 0.1493)(\frac{1}{42} + \frac{1}{106})}} = 0.513 \). The \( p \)-value is 0.607, which is not significant.

03

Verify Proportions with Larger Sample (Part b)

With larger samples, the proportions remain the same: \( \frac{210}{1260} = 0.1667 \) and \( \frac{450}{3180} = 0.1415 \). This confirms the proportions from part (a) with the new sample sizes.

04

Perform the z Test with Larger Sample

Recalculate the pooled proportion: \( \hat{p} = \frac{210 + 450}{1260 + 3180} = 0.1493 \). Compute the z statistic: \( z = \frac{0.1667 - 0.1415}{\sqrt{0.1493 (1 - 0.1493)(\frac{1}{1260} + \frac{1}{3180})}} = 2.816 \). The \( p \)-value < 0.05, indicating significance.

05

Calculate Confidence Interval for Part a

Using the sample proportions and sizes, calculate the standard error: \( SE = \sqrt{\frac{0.1667(1 - 0.1667)}{42} + \frac{0.1415(1 - 0.1415)}{106}} = 0.0564 \). The 95% confidence interval is \( 0.1667 - 0.1415 \pm 1.96 \times 0.0564 = (-0.0340, 0.0867) \).

06

Calculate Confidence Interval for Part b

For the larger sample, calculate the standard error: \( SE = \sqrt{\frac{0.1667(1 - 0.1667)}{1260} + \frac{0.1415(1 - 0.1415)}{3180}} = 0.0176 \). The 95% confidence interval is \( 0.1667 - 0.1415 \pm 1.96 \times 0.0176 = (0.0071, 0.0481) \).

07

Effect of Sample Size on Confidence Interval

Larger samples result in a narrower confidence interval, indicating more precision in estimating the difference between proportions.

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Key Concepts

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

z Test

Performing a z test is a statistical method used to determine if there is a significant difference between the proportions of two groups. In this exercise, we're comparing the failure rates of businesses headed by women and men. The test uses the null hypothesis, which suggests that there are no differences between the two groups (\( p_1 = p_2 \)).

The z test involves a calculation called the z statistic, which can tell us how far apart our actual observations are from the null hypothesis in terms of standard errors. To obtain this value, we calculate a quantity known as the pooled proportion, which estimates the common proportion under the null hypothesis:\[ \hat{p} = \frac{7 + 15}{42 + 106} = 0.1493 \]. Based on calculations provided in the solution, we find the initial z value to be 0.513. This statistic is then converted into a p-value to interpret significance. In this case, the p-value was 0.607, indicating that the observed difference between women's and men's business failures under the initial sample was statistically insignificant.

This changes dramatically with the larger sample in part (b). The recalculated z statistic of 2.816 pointed a clear \( p \)-value below 0.05, suggesting the difference in failure rates is statistically significant. This demonstrates how the effect size is easier to detect with larger sample sizes.

Confidence Interval

Confidence intervals provide a range of plausible values for an unknown parameter, in this scenario the difference in proportions of failures between women-led and men-led businesses. A 95% confidence interval (CI) means if we were to take 100 different samples and compute a CI for each, we'd expect around 95 of those intervals to contain the true difference in proportion.

In part (a), for the smaller sample size, the standard error (SE) is computed as follows:\[ \text{SE} = \sqrt{\frac{0.1667(1 - 0.1667)}{42} + \frac{0.1415(1 - 0.1415)}{106}} = 0.0564 \]. This leads to a 95% CI from -0.0340 to 0.0867, suggesting the real difference in failures might be between these bounds. However, this range includes zero, indicating non-significance.

For the larger sample size, the SE calculation gets more precise:\[ \text{SE} = \sqrt{\frac{0.1667(1 - 0.1667)}{1260} + \frac{0.1415(1 - 0.1415)}{3180}} = 0.0176 \], with a CI of 0.0071 to 0.0481. This narrower range doesn't include zero, aligning with the statistical significance observed, reinforcing how larger samples result in tighter, more informative intervals.

Proportion

A proportion is a type of ratio where we consider how many elements out of a total fall into a specific category. In everyday terms, if you have a proportion, you essentially have a fraction that represents a part of a whole. Here, the term refers to the share of businesses that failed: 7 out of 42 women-led and 15 out of 106 men-led businesses, initially yielding similar proportions of around 0.1667 and 0.1415 respectively.

Importantly, proportions are crucial in understanding the relative scale of an occurrence in different sized samples. For instance, when the sample size is increased 30 times (part b) to 1260 and 3180 businesses, the proportion of failures remains the same. This illustrates that while raw counts can change with sample size, the proportion as a measure of relative frequency often provides more insightful comparisons between groups.

Furthermore, understanding proportions is vital in hypothesis testing; assumptions such as those employed in z tests start with examining if two proportions differ significantly. Thus, comprehending the role of proportions helps in designing experiments and dissecting their outputs accurately.