Answer:
a) The 95% CI for the mean is
.
The value 274 is plausible because it is within the limits of the CI.
b) We can not reject the hypothesis ![H_0: \mu=274](https://tex.z-dn.net/?f=H_0%3A%20%5Cmu%3D274)
c) They are consistent. The conclusion from both results is that 274 could be the real mean. In both cases we couldn't prove 274 it is not the mean.
Step-by-step explanation:
In this problem we have a sample of n=32 with mean M=266.8 and standard deviation s=22.
a) To compute a 95% confidence interval (CI), we calculate
![M-t_{31}*s/\sqrt{n}\leq \mu\leq M+t_{31}*s/\sqrt{n}](https://tex.z-dn.net/?f=M-t_%7B31%7D%2As%2F%5Csqrt%7Bn%7D%5Cleq%20%5Cmu%5Cleq%20M%2Bt_%7B31%7D%2As%2F%5Csqrt%7Bn%7D)
First we have to estimate t, with 32-1=31 degrees of freedom for a two-tailed 95% CI.
By looking up in the t-table, we get t=2.0395.
Then the confidence interval is
![M-t_{31}*s/\sqrt{n}\leq \mu\leq M+t_{31}*s/\sqrt{n}\\\\266.8-2.0.395*22/\sqrt{32}\leq\mu\leq 266.8+2.0.395*22/\sqrt{32}\\\\266.8-7.9\leq\mu\leq 266.8+7.9\\\\258.8\leq\mu\leq 274.7](https://tex.z-dn.net/?f=M-t_%7B31%7D%2As%2F%5Csqrt%7Bn%7D%5Cleq%20%5Cmu%5Cleq%20M%2Bt_%7B31%7D%2As%2F%5Csqrt%7Bn%7D%5C%5C%5C%5C266.8-2.0.395%2A22%2F%5Csqrt%7B32%7D%5Cleq%5Cmu%5Cleq%20266.8%2B2.0.395%2A22%2F%5Csqrt%7B32%7D%5C%5C%5C%5C266.8-7.9%5Cleq%5Cmu%5Cleq%20266.8%2B7.9%5C%5C%5C%5C258.8%5Cleq%5Cmu%5Cleq%20274.7)
b) We have to test the hypothesis
![H_0: \mu=274\\\\H_1: \mu \neq 274](https://tex.z-dn.net/?f=H_0%3A%20%5Cmu%3D274%5C%5C%5C%5CH_1%3A%20%5Cmu%20%5Cneq%20274)
The significance level is 0.05.
We calculate the t-statistics:
![t=\frac{M-\mu}{s/\sqrt{n}}=\frac{266.8-274}{22/\sqrt{32}}=\frac{-7.2}{3.9}= 1.84](https://tex.z-dn.net/?f=t%3D%5Cfrac%7BM-%5Cmu%7D%7Bs%2F%5Csqrt%7Bn%7D%7D%3D%5Cfrac%7B266.8-274%7D%7B22%2F%5Csqrt%7B32%7D%7D%3D%5Cfrac%7B-7.2%7D%7B3.9%7D%3D%201.84)
The p-value for t=1.84 and df=31 is P=0.07536.
As P=0.07 is greater than the significance level (0.05), we can not reject the null hypothesis. The interpretation of this is we can not claim that the mean is not 274.