Question

1. according to the cdc (2016), the average life expectan - cy of someone with diabetes is $mu=72$ years, $sigma = 14$. suppose that a sample of $n = 64$ people diagnosed with diabetes who received a blood glucose monitor - ing implant had an average life expectancy of $m = 76$ years. test the hypothesis that the glucose monitoring implant changes life expectancy. assume a two - tailed test, $alpha=.05$.

Explanation:

Step1: Calculate the standard error

The formula for the standard error ($\sigma_M$) is $\sigma_M=\frac{\sigma}{\sqrt{n}}$.
Given $\sigma = 14$ and $n = 64$, then $\sigma_M=\frac{14}{\sqrt{64}}=\frac{14}{8}=1.75$.

Step2: Calculate the z - statistic

The formula for the z - statistic is $z=\frac{M-\mu}{\sigma_M}$.
Given $M = 76$, $\mu=72$ and $\sigma_M = 1.75$, then $z=\frac{76 - 72}{1.75}=\frac{4}{1.75}\approx2.29$.

Step3: Determine the critical z - values

For a two - tailed test with $\alpha = 0.05$, the critical z - values are $z=\pm1.96$.

Step4: Make a decision

Since $|z|=2.29>1.96$, we reject the null hypothesis.

Answer:

We reject the null hypothesis. The glucose monitoring implant changes life expectancy.