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Chapter 8: Q7CRE (page 356)

Lightning Deaths The graph in Cumulative Review Exercise 5 was created by using data consisting of 232 male deaths from lightning strikes and 55 female deaths from lightning strikes. Assume that these data are randomly selected lightning deaths and proceed to construct a 95% confidence interval estimate of the proportion of males among all lightning deaths. Based on the result, does it seem feasible that males and females have equal chances of being killed by lightning?

Short Answer

Expert verified

The 95% confidence interval estimate of the proportion of male deaths is equal to (0.762, 0.854).

Male and females do not have an equal chance of being killed by lightning.

Step by step solution

01

Given information

There are 232 male deaths and 55 female deaths due to lightning strikes.

02

Confidence Interval

The following formula is utilized to compute the confidence interval:

\(CI = \left( {\hat p - E,\hat p + E} \right)\)

Here,\(\hat p\)is the sample proportion of male deaths and E is the margin of error.

The margin of error has the following expression:

\(E = {z_{\frac{\alpha }{2}}}\sqrt {\frac{{\hat p\hat q}}{n}} \)

Here, n is the sample size,\(\hat p\)is the sample proportion of male deaths, and\(\hat q\)is the sample proportion of female deaths

\({z_{\frac{\alpha }{2}}}\) is the corresponding value of the standard normal distribution

03

Sample size and sample proportions

The sample size is computed below:

\(\begin{array}{c}n = 232 + 55\\ = 287\end{array}\)

The sample proportion of male deaths is equal to:

\(\begin{array}{c}\hat p = \frac{{{\rm{Number}}\;{\rm{of}}\;{\rm{male}}\;{\rm{deaths}}}}{{{\rm{Sample}}\;{\rm{size}}}}\\ = 0.808\end{array}\)

The sample proportion of female deaths is equal to:

\(\begin{array}{c}\hat q = 1 - \hat p\\ = 1 - 0.808\\ = 0.192\end{array}\)

04

Step 4:Value of \({z_{\frac{\alpha }{2}}}\)

The confidence level is equal to 95%. Thus, the corresponding level of significance\(\left( \alpha \right)\)is equal to 0.05.

The two-tailed value of \({z_{\frac{\alpha }{2}}}\) is equal to 1.96.

05

Computation of confidence interval

The value of the margin of error is equal to:

\(\begin{array}{c}E = {z_{\frac{\alpha }{2}}}\sqrt {\frac{{\hat p\hat q}}{n}} \\ = \left( {1.96} \right)\sqrt {\frac{{\left( {0.808} \right)\left( {0.192} \right)}}{{287}}} \\ = 0.0456\end{array}\)

The 95% confidence interval is equal to:

\(\begin{array}{c}CI = \left( {\hat p - E,\hat p + E} \right)\\ = \left( {0.808 - 0.0456,0.808 + 0.0456} \right)\\ = \left( {0.762,0.854} \right)\end{array}\)

Thus, the 95% confidence interval estimate of the proportion of male deaths is equal to (0.762,0.854).

Since the value of 0.5 is not included in the interval and the interval begins from 0.762, it can be inferred that the proportion of male and female deaths can never be the same and have a value equal to 0.5.

Thus, males and females do not have an equal chance of being killed by lightning.

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