Question

A random sample of 100 observations from a quantitative population produced a sample mean of 22.8 and a sample standard deviation of 8.3.

Use the p-value approach to determine whether the population mean is different from 24.

Explain your conclusions. (Use α = 0.05.)

State the null and alternative hypotheses. H0: μ = 24 versus Ha: μ ≠ 24 H0: μ = 24 versus Ha: μ > 24 H0: μ ≠ 24 versus Ha: μ = 24 H0: μ = 24 versus Ha: μ < 24 H0: μ < 24 versus Ha: μ > 24

Find the test statistic and the p-value. (Round your test statistic to two decimal places and your p-value to four decimal places.) z = p-value = State your conclusion.

Answer 1

We conclude that the population mean is 24.

Explanation:

We are given the following in the question:

Population mean, μ = 24

Sample mean,
`\bar{x}` = 22.8

Sample size, n = 100

Alpha, α = 0.05

Sample standard deviation, s = 8.3

First, we design the null and the alternate hypothesis

`H_(0): \mu = 24\\H_A: \mu \\eq 24`

We use Two-tailed z test to perform this hypothesis.

Formula:

`z_(stat) = \displaystyle\frac{\bar{x} - \mu}{(s)/(√(n)) }`

Putting all the values, we have

`z_(stat) = \displaystyle(22.8 - 24)/((8.3)/(√(100)) ) = -1.44`

We calculate the p-value with the help of standard z table.

P-value = 0.1498

Since the p-value is greater than the significance level, we accept the null hypothesis. The population mean is 24.

Now,
`z_(critical) \text{ at 0.05 level of significance } = \pm 1.96`

Since, the z-statistic lies in the acceptance region which is from -1.96 to +1.96, we accept the null hypothesis and conclude that the population mean is 24.