Respuesta :
Answer:
The null hypothesis is rejected. There is evidence that the proportion of adults who prefer mint chocolate chip ice cream differs from 25%.
The mean of the sampling distribution of the sample proportion is 0.25 and the standard deviation is 0.0242 (assuming the null hypothesis is true).
Step-by-step explanation:
We have the null and alternative hypothesis
[tex]H_0: \pi=0.25\\\\H_a:\pi\neq 0.25[/tex]
The level of significance is [tex]\alpha=0.05[/tex].
The sample mean is [tex]p=0.3[/tex]
The standard deviation is estimated as:
[tex]\sigma_p=\sqrt{\frac{\pi(1-\pi)}{N}}=\sqrt{\frac{0.25*0.75}{320}}=0.0242[/tex]
Then, the z-statistic can be calculated as:
[tex]z=\frac{p-\pi-0.5/N}{\sigma_p} =\frac{0.3-0.25+0.00}{0.0242} =\frac{0.05}{0.0242} \\\\z=2.07[/tex]
The P-value for this z=2.07 is
[tex]P-value=2*P(z>2.07)=0.04[/tex]
As the P-value is smallet than the significance level, the null hypothesis is rejected. There is evidence that the proportion of adults who prefer mint chocolate chip ice cream differs from 25%.
The sampling distribution will have a mean proportion equal to the population proportion (0.25), as it is not biased.
The standard deviation is calculated as before, and equals 0.0242.
[tex]\sigma_p=\sqrt{\frac{\pi(1-\pi)}{N}}=\sqrt{\frac{0.25*0.75}{320}}=0.0242[/tex]
