Respuesta :
Answer:
a) The null and alternative hypothesis are:
[tex]H_0: \pi=0.53\\\\H_a:\pi<0.53[/tex]
b) If 300 families were sampled, for a significance level of 5%, there is enough evidence to support the claim that a smaller proportion of American families own stocks or stock funds this year than 10 years ago (P-value = 0.001).
Step-by-step explanation:
The claim that we want to have evidence to support is that a smaller proportion of American families own stocks or stock funds this year than 10 years ago.
The hypothesis for this test should state:
- For the null hypothesis, that the population proportion is not significantly different from 53%.
[tex]H_0:\pi=0.53[/tex]
- For the alternative hypothesis, that the population proportion is significantly less than 53%.
[tex]H_a: \pi<0.53[/tex]
If 300 families are sampled, we can perform a hypothesis test for a proportion.
The claim is that a smaller proportion of American families own stocks or stock funds this year than 10 years ago.
Then, the null and alternative hypothesis are:
[tex]H_0: \pi=0.53\\\\H_a:\pi<0.53[/tex]
The significance level is 0.05.
The sample has a size n=300.
The sample proportion is p=0.44.
The standard error of the proportion is:
[tex]\sigma_p=\sqrt{\dfrac{\pi(1-\pi)}{n}}=\sqrt{\dfrac{0.53*0.47}{300}}\\\\\\ \sigma_p=\sqrt{0.00083}=0.029[/tex]
Then, we can calculate the z-statistic as:
[tex]z=\dfrac{p-\pi+0.5/n}{\sigma_p}=\dfrac{0.44-0.53+0.5/300}{0.029}=\dfrac{-0.088}{0.029}=-3.065[/tex]
This test is a left-tailed test, so the P-value for this test is calculated as:
[tex]\text{P-value}=P(z<-3.065)=0.001[/tex]
As the P-value (0.001) is smaller than the significance level (0.05), the effect is significant.
The null hypothesis is rejected.
There is enough evidence to support the claim that a smaller proportion of American families own stocks or stock funds this year than 10 years ago.
