Question

A large survey of countries including the USA, China, Russia, France, and others indicated that most people prefer the color blue. In fact, about 24% of the population claim blue as their favorite color. Suppose a random sample of 86 college students were surveyed and 30 of them said that blue is their favorite color. Does this information imply that the color preference of all college students is different (either way) from that of the general population? Use α = 0.05.1. State your null and alternate hypothesis. What is the level of significance? Will you use a left tail, right tail or two-tail test?2. What is the value of the test statistic. 3.lFind the P-value. Sketch the sampling distribution z = to show the area corresponding to the P-value. 4. Based on 1-3, will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α ? State your conclusion in the context of the application.

Answer 1

We have to perform an hypothesis test of a proportion.

The claim is that the sample has a different proportion than the population.

Then, the null and alternative hypothesis are:

`\begin{gathered} H_0\colon\pi=0.24 \\ H_a\colon\pi\\eq0.24 \end{gathered}`

The significance level is 0.05.

The sample has a size n=86.

The sample proportion is p=0.349.

`p=X/n=30/86=0.349`

The standard error of the proportion is:

`\begin{gathered} \sigma_p=\sqrt{(\pi(1-\pi))/(n)}=\sqrt[]{(0.24\cdot0.76)/(86)} \\ \sigma_p=√(0.002121)=0.046 \end{gathered}`

Then, we can calculate the z-statistic as:

`z=(p-\pi-0.5/n)/(\sigma_p)=(0.349-0.24-0.5/86)/(0.046)=(0.103)/(0.046)=2.241`

This test is a two-tailed test*, so the P-value for this test is calculated as:

`\text{P-value}=2\cdot P(z>2.241)=0.025`

* We use a two-tailed test because we are looking for difference above or below the population proportion.

As the P-value (0.025) is smaller than the significance level (0.05), the effect is significant.

The null hypothesis is rejected.

At a significance level of 0.05, there is enough evidence to support the claim that the sample has a different proportion than the population.

Answer:

1) The null and alternative hypothesis are:

`\begin{gathered} H_0\colon\pi=0.24 \\ H_a\colon\pi\\eq0.24 \end{gathered}`

2) The test statistic is z=2.241.

3) The P-value is 0.025. The value in the standard normal distribution is:

4) As the effect is significant (the P-value is less than the significance level), there is evidence to reject the null hypothesis.

The conclusion is that this sample has a proportion that is significantly different from that from the population.