Question

Hypothesis Test for Lightning Deaths Refer to the sample data given in Cumulative Review Exercise 1 and consider those data to be a random sample of annual lightning deaths from recent years. Use those data with a 0.01 significance level to test the claim that the mean number of annual lightning deaths is less than the mean of 72.6 deaths from the 1980s. If the mean is now lower than in the past, identify one of the several factors that could explain the decline.

Short answer

There is enough evidence to support the claim that the mean number of deaths due to lightning strikes is less than 72.6.

By practicing better lightning protection techniques with the help of advanced tools and equipment such as lightning arrestors, the number of deaths due to lightning has declined.

Step-by-step solution

Given information

Data are given on the number of deaths that have occurred due to lightning strikes for a series of 14 consecutive years. It is claimed that the mean number of deaths is less than 72.6.

Hypotheses

Corresponding to the given claim, the following hypotheses are set up:

Null Hypothesis: The mean number of deaths is equal to 72.6.

$H_0:\mu =72.6$

Alternative Hypothesis: The mean number of deaths is less than 72.6.

$H_1:\mu <72.6$

The test is left-tailed.

Important values

The sample size (n) is equal to.

The sample mean is computed below:

$$\begin{gathered} \overline{x}=\frac{51+44+........+23}{14} \\ =37.1 \end{gathered}$$

The sample standard deviation is computed below:

$$\begin{gathered} s=\sqrt{\frac{{\sum }_{i=1}^n{(x_i-\overline{x})}^2}{n-1}} \\ =\sqrt{\frac{{\left(51-37.1\right)}^2+{\left(44-37.1\right)}^2+.......+{\left(23-37.1\right)}^2}{14-1}} \\ =9.8 \end{gathered}$$

The population mean is equal to 72.6.

Test statistic

The test statistic value is obtained as follows:

$$\begin{gathered} t=\frac{\overline{x}-\mu }{\frac{s}{\sqrt{n}}} \\ =\frac{37.1-72.6}{\frac{9.8}{\sqrt{14}}} \\ =-13.55 \end{gathered}$$

Thus, t=-13.55.

The degrees of freedom are computed below:

$$\begin{gathered} df=n-1 \\ =14-1 \\ =13 \end{gathered}$$

Referring to the student’s t distribution table, the critical value of t at$\alpha =0.05$ and 13 degrees of freedom for a left-tailed test is equal to -1.7709. The p-value corresponding to the test statistic value of -13.51 is equal to 0.000.

Since the p-value is less than 0.05, the null hypothesis is rejected.

Conclusion

There is enough evidence to support the claim that the mean number of deaths due to lightning strikes is less than 72.6.

One possible factor due to which there is a significant decline in the number of deaths caused by lightning is that the infrastructure built is suitable for the prevention of casualties by proper grounding of electric cables and phone lines and using better equipment such as lightning arrestors.