Answer:
The probability that we fail to reject H0 when H0 is false is P=0.48.
Step-by-step explanation:
Failing to reject the null hypothesis when it is false is a Type II error.
The probability of a Type II error is written as β.
We have a null hypothesis that states that the mean is 99 degrees F. The standard deviation of the population is known and is 2 degrees F.
We assume a significance level of 0.05 and a two-tails test.
The z-value for this level of significance is z=1.96.
The value of β can be calculated as:
[tex]\beta=\Phi(z_{\alpha}-\frac{\delta}{\sigma/\sqrt{n}})[/tex]
Where Φ is the standarized normal function and δ is the difference between the real mean and the null hypothesis mean.
Then, δ is:
[tex]\delta=\mu-\mu_0=100-99=1[/tex]
Now, we can calculate β as:
[tex]\beta=\Phi(z_{\alpha}-\frac{\delta}{\sigma/\sqrt{n}})\\\\\beta=\Phi(1.96-1/(2/\sqrt{16}))\\\\\beta=\Phi(1.96-2)=\Phi(-0.04)=0.484[/tex]
