Question

Employees of a large corporation are concerned about the declining quality of medical services provided by their group health insurance. A random sample of 100 office visits by employees of this corporation to primary care physicians during the year 2017 found that the doctors spent an average of 19 minutes with each patient. This year a random sample of 108 such visits showed that doctors spent an average of 15.5 minutes with each patient. Assume that the standard deviations for the two populations are 2.7 and 2.1 minutes, respectively. Using the 2.5% level of significance, can you conclude that the mean time spent by doctors with each patient is lower for this year than for 2017? To draw your conclusion, state the hypotheses and identify the claim, find the critical value(s), label the acceptance and rejection region, calculate the test value and summarize the results. 

Answer 1

The following null and alternative hypotheses need to be tested:

$H_0:\mu_1\leq \mu_2$

$H_1:\mu_1>\mu_2$

This corresponds to a right-tailed test, and a z-test for two means, with known population standard deviations will be used.

Based on the information provided, the significance level is $\alpha=0.025,$ and the critical value for a right-tailed test is $z_c=1.96.$

The rejection region for this right-tailed test is $R=\{z:z>1.96\}.$

The z-statistic is computed as follows:

$$z=\dfrac{\bar{x}_1-\bar{x}_2}{\sqrt{\sigma^2_1/n_1+\sigma_2^2/n_2}}$$

$$=\dfrac{19-15.5}{\sqrt{2.7^2/100+2.1^2/108}}\approx10.37825$$

Since it is observed that $z=10.37825>1.96=z_c,$ it is then concluded that the null hypothesis is rejected.

Therefore, there is enough evidence to claim that the population mean $\mu_1$ is greater than $\mu_2,$ at the $\alpha=0.025$ significance level.

Using the P-value approach: The p-value is $p=P(Z>10.37825)=0,$ and since $p=0<0.025=\alpha,$ it is concluded that the null hypothesis is rejected.

Therefore, there is enough evidence to claim that the population mean $\mu_1$ is greater than $\mu_2,$ at the $\alpha=0.025$ significance level.

Therefore, there is enough evidence to claim that the mean time spent by doctors with each patient is lower for this year than for 2017 at the $\alpha=0.025$ significance level.