Question

Research objective: The population mean is greater
than 7.5.
σ = 1.5, n = 30, x = 8.5

Answer 1

Null hypothesis H₀: μ $\leq$ 7.5

Alternative hypotheses H₁: μ > 7.5

The level of significance can be considered as 0.05.

The test statistic for one sample t test is:

$t = \frac{x - μ}{\frac{σ}{\sqrt{n}}} \\ = \frac{8.5 - 7.5}{\frac{1.5}{\sqrt{30}}} \\ = \frac{1.0}{0.274} \\ = 3.65$

The degrees of freedom can be calculated as:

n – 1 = 30 – 1

= 29

Determine the p-value of the test from standard normal table as below:

$P(Z>z) = 1-P(Z<z) \\ = 1 -P(Z< 3.65) \\ = 1 - 0.9998 \\ = 0.0002 ≈ 0$

The decision is to reject the null hypothesis if p-value is less than the considered level of significance. Here, p-value is less than the significance level. The researcher will reject the null hypothesis. Hence, there is sufficient evidence to say that the population mean is less than 7.5.