Chapter 11.2, Problem 10E

Pollution and altitude: In a random sample of 340 cars driven at low altitudes, 46 of them exceeded a standard of 10 grams of particulate pollution per gallon of fuel consumed. In an independent random sample of 85 cars driven at high altitudes, 21 of them exceeded the standard. Can you conclude that the proportion of high-altitude vehicles exceeding the standard is greater than the proportion of low-altitude vehicles exceeding the standard?

1. State the appropriate null and alternate hypotheses.
2. Compute the value of the test statistic.
3. State a conclusion at the $\alpha =0.01$ level of significance.

To determine

To find:

The appropriate null and alternate hypotheses.

Answer to Problem 10E

The null and alternative hypotheses are,

  $\begin{array}{l}{H}_0=p_1=p_2\\ H_a=p_1>p_2\end{array}$

Explanation of Solution

Given Information:

The sample size at low altitude is $340$ ,the number of cars exceeded a standard of $10$ grams of particulate pollution per gallon of fuel consumedis $46$ , the sample size at high altitude is $85$ and the number of cars exceeds the standard is $21$ .

Formula used:

The null and alternative hypotheses are,

  $\begin{array}{l}{H}_0=p_1=p_2\\ H_a=p_1>p_2\end{array}$

Here, $p_1$ is the proportion of population $1\text{ and }{p}_2$ is the proportion of population $2$ .

Calculation:

The null and alternative hypotheses are,

  $\begin{array}{l}{H}_0=p_1=p_2\\ H_a=p_1>p_2\end{array}$

To determine

To find:

The value of the test statistic.

Answer to Problem 10E

The z-test statistics is $z=2.53$ .

Explanation of Solution

Given Information:

The sample size at low altitude is $340$ ,the number of cars exceeded a standard of $10$ grams of particulate pollution per gallon of fuel consumedis $46$ , the sample size at high altitude is $85$ and the number of cars exceeds the standard is $21$ .

Formula used:

Test the hypothesis is that the proportion of high-altitude vehicles exceeding the standard is higher than the proportion of low-altitude vehicles exceeding. The standard at $1\%$ significance level.

Let the level of significance be $\alpha =0.01$ .

Using The Minitab follow the steps mentioned below:

  $1:$ Import the data.

  $2:$ Select $2$ Proportions from Basic Statistics from Stat menu.

  $3:$ Enter Summarized data.

  $4:$ Click option button choose level of significance and alternative hypothesis

  $5:$ Click OK

Calculation:

The obtained Minitab output for the difference of two proportions is,

Test and $\text{CI}$ for two proportions

  $\begin{array}{l}{p}_1:\text{ Proportion where Sample }1=\text{ Event}\\ p_2:\text{ Proportion where Sample 2}=\text{Event}\\ \text{Difference}:p_1-p_2\end{array}$

Descriptive Statistics

  $\frac{\text{Sample N Event Sample }p}{\begin{array}{l}\text{Sample 1 85 21 0}\text{.247059}\\ \text{Sample 2 340 46 0}\text{.135294}\end{array}}$

Estimation for Difference

  $\frac{\begin{array}{l}\text{99\% Lower}\\ \text{Bound for}\\ \text{Difference }\text{Difference }\end{array}}{0.111765-0.005308}$

  $\text{CI}$ Based on normal approximation.

Test

  $\frac{\begin{array}{l}\text{Null hypothesis }{\text{H}}_0:p_1-p_2=0\\ \text{Alternnative hypothesis}{\text{H}}_1:p_1-p_2>0\\ \text{Method Z-Value p-Value}\end{array}}{\begin{array}{l}\text{Normal approximation }\text{2}\text{.53}0.006\\ \text{Fisher's exact }\text{0}\text{.011}\end{array}}$

The pooled estimate of the proportion $\left(0.157647\right)$ is used for the tests.

From the obtained Minitab output, the z-test statistics is $z=2.53$ .

To determine

To find:

The conclusion at the given level of significance.

Answer to Problem 10E

From the obtained Minitab output, the right tailed P-value is $0.006$ . The P-value is less than the significance level of $0.01$ . Reject the null hypothesis.

Explanation of Solution

Given Information:

  $P-value=0.006$

Formula used:

.If the P-value is less than significance level of $0.01$ . Reject the null hypothesis.

Calculation:

From the obtained Minitab output, the right tailed P-value is $0.006$ . The P-value is less than the significance level of $0.01$ . Reject the null hypothesis.

Therefore, it can be concluded that there is a sufficient evidence to indicate that the proportion of high-altitude vehicles exceeding the standard is higher than the proportion of low-altitude vehicles exceeding the standard. The result is statistically significant.