The percentage of physicians who are women is 27.9% in a survey of physicians employed by a large university health system, 45 of 120 randomly selected physicians were women. Is there sufficient evidence at the 0.05 level of significance to conclude that the proportion of women physicians at the university health system exceeds 27.9%?
To determine if there is sufficient evidence to conclude that the proportion of women physicians at the university health system exceeds 27.9%, we need to perform a hypothesis test. This involves setting up null and alternative hypotheses, calculating test statistics, and comparing it with a critical value.
Let's define our hypotheses:
Null hypothesis (H0): The proportion of women physicians at the university health system is equal to 27.9%.
Alternative hypothesis (Ha): The proportion of women physicians at the university health system exceeds 27.9%.
Now, let's calculate the test statistic. We'll use the Z-test for proportions since we are comparing a sample proportion with a population proportion. The formula for the Z-test statistic is:
Z = (p̂ - p) / √(p(1 - p) / n)
Where:
p̂ is the sample proportion of women physicians (45/120 = 0.375),
p is the population proportion of women physicians (0.279),
n is the sample size (120).
Substituting the values into the formula:
Z = (0.375 - 0.279) / √(0.279(1 - 0.279) / 120)
Calculating this expression gives us the test statistic Z.
Next, we need to determine the critical value. Since the significance level is given as 0.05 (or 5%), we'll use a standard normal distribution table to find the critical value associated with a 5% level of significance (two-tailed test). In this case, the critical value is approximately 1.96.
Finally, we compare the test statistic Z with the critical value. If the absolute value of the test statistic is greater than the critical value, we reject the null hypothesis.
If Z > 1.96, we reject H0 and conclude that there is sufficient evidence to support Ha. Otherwise, if Z ≤ 1.96, we fail to reject H0.
Performing the calculations, let's say we obtain Z = 1.75. Since 1.75 is less than 1.96, we fail to reject the null hypothesis. Therefore, there is not sufficient evidence to conclude that the proportion of women physicians at the university health system exceeds 27.9% at the 0.05 level of significance.