Question

question 2
1 pts
a recent technology report claims that 55% of mid - sized companies use al tools for data analytics. a consulting firm believes this percentage has increased because of advances in generative al. 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?
if the original claim is true, results this far above 55% would occur about 2.87% of the time.
because the p - value is smaller than 0.05, there is less than a 5% chance the original claim is true.
the consulting firm did not compute the test statistic correctly.
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.
since the p - value is smaller than 0.05, this proves that 60% of all mid - sized companies now use al tools for data analytics.

Explanation:

Step1: Find the P - value for a one - tailed test

For a one - tailed z - test, if \(z = 1.9\), we look up the value in the standard normal table. The standard normal table gives the area to the left of \(z\). For \(z=1.9\), the area to the left \(P(Z < 1.9)\) is \(0.9713\). The area to the right (our P - value for a right - tailed test) is \(P(Z>1.9)=1 - P(Z < 1.9)\)

$$P(Z>1.9)=1 - 0.9713=0.0287 = 2.87\%$$

Step2: Analyze each option

• Option 1:

Since \(P(Z > 1.9)=2.87\%\), if the null hypothesis (the original claim \(p = 0.55\)) is true, results this far above \(55\%\) (in the right - tailed test) would occur about \(2.87\%\) of the time. This is correct.

• Option 2:

The P - value is the probability of getting the sample result (or more extreme) if the null hypothesis (original claim) is true, not the probability that the null hypothesis is true. So this option is wrong.

• Option 3:

We are not given any information to suggest the consulting firm miscalculated the \(z\) - statistic. So this option is wrong.

• Option 4:

A \(z\) - value of \(1.9\) with a P - value of \(2.87\%\) (less than \(5\%\)) leads us to reject the null hypothesis (original claim), not keep assuming it. So this option is wrong.

• Option 5:

Hypothesis testing does not prove the alternative hypothesis (\(p=0.6\) in this case). It just gives evidence against the null hypothesis. So this option is wrong.

Answer:

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