A state survey investigates whether the proportion of 8% for employees who commute by car to work is higher than it was five years ago. A test on employee commuting by car was done on a random sample size of 1000 and found 100 commuters by car. Test an appropriate hypothesis using α = 0.05.
A) P-value =0.0099. There is a high chance of having 100 or less of 1000 people in a random sample be commute by car if in fact 8% do.
B) P-value =0.0099. The change is statistically significant. A 98% confidence interval is (7.8%, 12.2%). This is clearly lower than 8%. The chance of observing 100 or more commuters by car of 1000 is 1% if the commuting by car is really 8%.
C) P-value =0.9901. There is a 99% chance of having 100 or less of 1000 people in a random sample be commute by car if in fact 8% do.
D) P-value =0.9901. The change is statistically significant. A 90% confidence interval is (8.4%, 11.6%). This is clearly higher than 8%. The chance of observing 100 or more commuters by car of 1000 is less than 99% if the commuting by car is really 8%.
E) P-value =0.0099. The change is statistically significant. A 90% confidence interval is (8.4%, 11.6%). This is clearly higher than 8%. The chance of observing 100 or more commuters by car of 1000 is 1% if the commuting by car is really 8%. The P-value is less than the alpha level of 0.05.
To test the hypothesis, we need to perform a hypothesis test for proportions.
Here are the steps for conducting the hypothesis test:
1. State the null hypothesis (H0) and alternative hypothesis (Ha):
- Null hypothesis (H0): The proportion of employees who commute by car is equal to 8%.
- Alternative hypothesis (Ha): The proportion of employees who commute by car is higher than 8%.
2. Determine the significance level (α): In this case, the significance level (α) is given as 0.05.
3. Collect the data and calculate the test statistic: From the given information, we have a random sample of 1000 employees, with 100 of them commuting by car. The test statistic, in this case, is the proportion of employees who commute by car in the sample.
4. Calculate the p-value: The p-value is the probability of observing a test statistic as extreme as or more extreme than the one calculated, assuming the null hypothesis is true.
5. Compare the p-value with the significance level (α): If the p-value is less than the significance level (α), we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
Now let's evaluate the given answer choices:
A) The p-value is 0.0099. This means that there is a low probability of observing 100 or fewer commuters by car out of 1000 in a random sample if the true proportion is 8%. However, the question asks for a higher proportion, so this option is incorrect.
B) The p-value is 0.0099, indicating that the change is statistically significant. Additionally, a 98% confidence interval is provided, which does not contain the value 8%. This suggests that the proportion is lower than 8%. The chance of observing 100 or more commuters by car out of 1000 is given as 1%, which is less than the significance level (α = 0.05). Therefore, this option is correct.
C) The p-value is given as 0.9901, which is contradictory to the information provided in option A. This option also does not mention anything about statistical significance or confidence intervals, so it is incorrect.
D) Similar to option C, this option also contradicts option B and does not provide information about the p-value. It only presents a different confidence interval and probability, which is not directly related to the hypothesis test. Therefore, this option is incorrect.
E) This option combines information from option B with additional information about the significance level (α = 0.05) and the p-value being less than the alpha level. Therefore, this option is correct.
Therefore, the correct answer is option E: P-value = 0.0099. The change is statistically significant. A 90% confidence interval is (8.4%, 11.6%). This is clearly higher than 8%. The chance of observing 100 or more commuters by car out of 1000 is 1% if the commuting by car is really 8%. The p-value is less than the alpha level of 0.05.