#12 **Statistical Analysis of Air Bag Vehicles in Fatal Auto Accidents****Problem Statement:**Researchers at VCU studied fatal auto accidents between cars in which one has an air bag and the other does not. They found that in 73% of all such fatal crashes, the driver of the air bag vehicle was responsible for the accident. Suppose that you obtain police records of fatal auto accidents. A random sample of 41 such accidents showed that 33 were the fault of the driver of the air bag vehicle. Does this indicate that the population proportion of such accidents in which the driver of the air bag vehicle was at fault is higher than 73%? Use a 5% level of significance.**Step-by-Step Solution:****a. State the hypotheses:**\[\text{H}_0: p = 0.73\] \[\text{H}_1: p > 0.73\]**b. Identify the level of significance and the critical value:**\[\alpha = 0.05\]\[z_0 = \text{from standard normal distribution for a one-tailed test at } \alpha = 0.05\]**c. Calculate the test statistic (z):**The test statistic is calculated using the formula:\[z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}}\]Where:\[\hat{p} = \frac{33}{41}\]\[p_0 = 0.73\]\[n = 41\]**d. Calculate the p-value:**\[P = \text{P(Z > z)} \] **e. Decision:**Compare the calculated p-value with the significance level (α):- If $P \leq \alpha$, reject the null hypothesis, H0.- If $P > \alpha$, fail to reject the null hypothesis, H0._z-Score Formula:_\[z = \frac{X - \mu}{\sigma}\]_or_\[z = \frac{\bar{X} - \mu}{\frac{\sigma}{\sqrt{n}}}\]_End of Problem_This statistical procedure helps determine whether the proportion of drivers at fault in fatal auto accidents involving air bag vehicles is statistically significantly higher than 73%.

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12. Researchers at VCU studied fatal auto accidents between cars in which one has an air bag and the other does not. They found that in 73% of all such fatal crashes the driver of the air bag vehicle was responsible for the accident. Suppose that you obtain police records of fatal auto accidents. A random sample of 41 such accidents showed that 33 were the fault of the driver of the air bag vehicle. Does this indicate that the population proportion of such accidents in which the driver of the air bag vehicle was at fault is higher than 73%? Use a 5% level of significance.

12. Researchers at VCU studied fatal auto accidents between cars in which one has an air bag and the other does not. They found that in 73% of all such fatal crashes the driver of the air bag vehicle was responsible for the accident. Suppose that you obtain police records of fatal auto accidents. A random sample of 41 such accidents showed that 33 were the fault of the driver of the air bag vehicle. Does this indicate that the population proportion of such accidents in which the driver of the air bag vehicle was at fault is higher than 73%? Use a 5% level of significance.

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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Question

#12

**Statistical Analysis of Air Bag Vehicles in Fatal Auto Accidents**

**Problem Statement:**

Researchers at VCU studied fatal auto accidents between cars in which one has an air bag and the other does not. They found that in 73% of all such fatal crashes, the driver of the air bag vehicle was responsible for the accident. Suppose that you obtain police records of fatal auto accidents. A random sample of 41 such accidents showed that 33 were the fault of the driver of the air bag vehicle. Does this indicate that the population proportion of such accidents in which the driver of the air bag vehicle was at fault is higher than 73%? Use a 5% level of significance.

**Step-by-Step Solution:**

**a. State the hypotheses:**
\[\text{H}_0: p = 0.73\]
\[\text{H}_1: p > 0.73\]

**b. Identify the level of significance and the critical value:**
\[\alpha = 0.05\]
\[z_0 = \text{from standard normal distribution for a one-tailed test at } \alpha = 0.05\]

**c. Calculate the test statistic (z):**

The test statistic is calculated using the formula:
\[z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}}\]

Where:
\[\hat{p} = \frac{33}{41}\]
\[p_0 = 0.73\]
\[n = 41\]

**d. Calculate the p-value:**
\[P = \text{P(Z > z)} \]

**e. Decision:**

Compare the calculated p-value with the significance level (α):

- If $P \leq \alpha$, reject the null hypothesis, H0.
- If $P > \alpha$, fail to reject the null hypothesis, H0.

_z-Score Formula:_
\[z = \frac{X - \mu}{\sigma}\]
_or_
\[z = \frac{\bar{X} - \mu}{\frac{\sigma}{\sqrt{n}}}\]

_End of Problem_

This statistical procedure helps determine whether the proportion of drivers at fault in fatal auto accidents involving air bag vehicles is statistically significantly higher than 73%.

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