Question

Use a​ t-test to test the claim about the population mean at the given level of significance using the given sample statistics. Assume the population is normally distributed.

Claim: μ ≥8 300, α = 0.10
Sample statistics: x = 8000, s = 440, n = 24
A. What are the null and alternative hypotheses?
B. What is the value of the standardized test statistic?
C. What is the p-value?
D. Decide whether to reject or fail to reject the null hypothesis.

Answer 1

A

The null hypothesis is
`H_o : \mu \ge 8300`

The alternative hypothesis is
`H_a : \mu < 8300`

B

`t = -3.34`

C

`p-value = P(t< -3.34) = 0.00041889`

D

reject the null hypothesis

Explanation:

From the question we are told that

The population mean is
`\mu = 8300`

The sample mean is
`\ = x = 8000`

The standard deviation is
`s = 440`

The sample size is
`n = 24`

The level of significance is
`\alpha = 0.01`

The null hypothesis is
`H_o : \mu \ge 8300`

The alternative hypothesis is
`H_a : \mu < 8300`

The test statistic is mathematically evaluated as

`t = (\= x - \mu )/( (s)/(√(n) ) )`

=>
`t = (8000- 8300 )/( (440)/(√(24) ) )`

=>
`t = -3.34`

The p-value is obtained from the z -table ( reference calculator dot net ) , the value is

`p-value = P(t< -3.34) = 0.00041889`

Looking at the values of
`p-value and \ \alpha` we see that
`p-value < \alpha` Hence we reject the null hypothesis