Answer :
To test the claim that at least 80% of doctors recommend aspirin for headaches, we will perform a hypothesis test for a proportion.
Step 1: Define the Hypotheses
- Null Hypothesis ([tex]H_0[/tex]): [tex]p \geq 0.80[/tex]
- Alternative Hypothesis ([tex]H_a[/tex]): [tex]p < 0.80[/tex]
Where [tex]p[/tex] is the true proportion of doctors who recommend aspirin.
Step 2: Calculate the Test Statistic
The formula for the test statistic for a proportion is:
[tex]z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}[/tex]
Where:
[tex]\hat{p}[/tex] is the sample proportion (72 out of 100 doctors), calculated as [tex]\hat{p} = \frac{72}{100} = 0.72[/tex].
[tex]p_0[/tex] is the hypothesized population proportion, which is 0.80.
[tex]n[/tex] is the sample size, which is 100.
Substitute the values into the formula:
[tex]z = \frac{0.72 - 0.80}{\sqrt{\frac{0.80 \times 0.20}{100}}}[/tex]
[tex]z = \frac{-0.08}{\sqrt{\frac{0.16}{100}}}[/tex]
[tex]z = \frac{-0.08}{0.04}[/tex]
[tex]z = -2.00[/tex]
Step 3: Compare the Test Statistic to the Critical Value
Since this is a left-tailed test and [tex]\alpha = 0.05[/tex], we compare the calculated [tex]z[/tex] to the critical value of [tex]z[/tex] at the 0.05 significance level (which is -1.645 for a one-tailed test).
Conclusion:
The calculated test statistic is [tex]z = -2.00[/tex]. Since [tex]-2.00 < -1.645[/tex], we reject the null hypothesis.
This means there is sufficient evidence to suggest that less than 80% of doctors recommend aspirin for their patients with headaches.
Thus, the multiple choice answer is -2.00.
