Question

1. a recent technology report claims that 55% of mid - sized companies use ai tools for data analytics. a consulting firm believes this percentage has increased because of advances in generative ai. the firm surveys 150 mid - sized companies and obtains a sample proportion of 0.60. when a hypothesis test is conducted, the consulting firm computes a value of z equal to 1.9. based on this information, what should be concluded?

a. if the original claim is true, results this far above 55% would occur about 2.87% of the time.
b. because the p - value is smaller than 0.05, there is less than a 5% chance the original claim is true.
c. the consulting firm did not compute the test statistic correctly.
d. the value of z = 1.9 shows the data are close enough to the original claim that we should keep assuming that claim is true.
e. since the p - value is smaller than 0.05, this proves that 60% of all mid - sized companies now use ai tools for data analytics.

Explanation:

Step1: Calculate the P - value

For a one - sided z - test (since we are testing if the proportion has increased), the P - value is \(P(Z>1.9)\).
Using the standard normal distribution table, \(P(Z > 1.9)=1 - P(Z\leq1.9)\).
From the standard normal table, \(P(Z\leq1.9)=0.9713\).
So, \(P(Z > 1.9)=1 - 0.9713 = 0.0287=2.87\%\).

Step2: Analyze each option

• Option A: The P - value of \(0.0287\) (or \(2.87\%\)) represents the probability of getting a result as extreme or more extreme than the observed sample result, assuming the null hypothesis (original claim) is true. This option is correct.
• Option B: The P - value is the probability of getting the sample result (or more extreme) if the null hypothesis is true, not the probability that the null hypothesis is true. So this option is incorrect.
• Option C: There is no indication that the z - value calculation is wrong. So this option is incorrect.
• Option D: A z - value of \(1.9\) with a P - value of \(0.0287<0.05\) (a common significance level) means we reject the null hypothesis (original claim), not keep assuming it. So this option is incorrect.
• Option E: Hypothesis testing does not prove a claim. It provides evidence against or in favor of the null hypothesis. So this option is incorrect.

Answer:

A. If the original claim is true, results this far above 55% would occur about 2.87% of the time.