A random sample of 10 college students has a mean earnings of $3120 with a sample standard deviation of $677 over the summer months. Determine whether a normal distribution or a t-distribution should be used or whether neither of these can be used to test a claim: μ > $3000.

We are given that a random sample of 10 college students has a mean earnings of $3,120 with a sample standard deviation of $677 over the summer months.

We have to test a claim of μ > $3,000.

Since in this question we are provided with;

Sample mean earnings, [tex]\bar X[/tex] = $3,120

Sample standard deviation, s = $677

Sample of college students, n = 10

So, Null Hypothesis, [tex]H_0[/tex] : [tex]\mu \leq[/tex] $3,000

Alternate Hypothesis, [tex]H_a[/tex] : [tex]\mu[/tex] > $3,000

The distribution that we will use here for our test statistics will be t-distribution because in the question we don't know anything about population standard deviation [tex](\sigma)[/tex] .

Normal distribution is used when we know population standard deviation [tex](\sigma)[/tex] .

So, the test statistics used will be One-sample t-test statistics;

Test statistics = [tex]\frac{\bar X -\mu}{\frac{s}{\sqrt{n} } }[/tex] ~ [tex]t_n_-_1[/tex]

Therefore, t-distribution should be used to test a claim: μ > $3000.