Question

At one college, the proportion of students that needed to go into debt to buy their supplies each term (books, tech, etc.) was once known to be 74%. An SRS of 60 students was then surveyed in a later term, in order to see if this previous proportion would still be supported by the new sample evidence.

Out of these 60 sampled students, 52 needed to go into debt to buy their supplies that term. Using a normal distribution of approximation (according to the CLT): We will conduct a two-sided significance test at a 3% significance level, to see if our sample has produced statistically significant evidence.

1. State the hypotheses (both the null and alternative) for this two-sided significance test.

2. Using N(μ, σ) normal distribution notation: Identify the very specific normal distribution (by stating the exact numerical values for μ and σ within this notation) that should be used to perform this test.

3. Find the percent P-value from this test.

Answer 1

Let p be the true proportion of students that needed to go into debt to buy their supplies in the term of the survey.

We want to test if p is still equal to 74%.

The hypotheses are

The sample information is

n=60 is the sample size

is the sample proportion of students that are in debt

is the hypothesized value of proportion (using the null hypothesis)

The standard error of proportion is

Since both are greater than 5, we can use CLT to use a normal approximation for the distribution of .

That is the sampling distribution of is normal with mean and the standard deviation

ans: needs to be used to perform this test.

The test statistics is

This is a 2 tailed test (The alternative hypothesis has "not equal to")

The p-value is

The p-value for this test is 0.025 (2.5% in percent terms)

We will reject the null hypothesis if the p-value is less than the significance level 3%.

Here, the p-value is 2.5% and it is less than 3%. Hence we reject the null hypothesis.

We conclude that at 3% significance, there is sufficient evidence to support the claim that the proportion of students that need to go into debt to buy their supplies is different from 74%  that term.