Answer
We fail to reject the null hypothesis.
Work Step by Step
Step 1: State the null and alternative hypotheses. The null hypothesis (H₀) is that the population mean is greater than or equal to $51$, and the alternative hypothesis (H₁) is that the population mean is less than $51$.
$H₀: μ ≥ 51$
$ H₁: μ < 51$
Step 2:
Determine the level of significance. The level of significance, $α$, is given as $0.01$.
Step 3:
Calculate the test statistic. We will use the t-test statistic since the population standard deviation is unknown. The formula for the t-test statistic is:
\[ t = \frac{{\overline{x} - \mu}}{{s / \sqrt{n}}} \]
where $x̄$ is the sample mean, $μ$ is the population mean, s is the sample standard deviation, and n is the sample size.
\[ t = \frac{{52 - 51}}{{\frac{{2.5}}{{\sqrt{40}}}}} \approx 2.5298 \]
Step 4:
Determine the critical value. Since this is a left-tailed test (H₁: μ < 51) with $α = 0.01$ and degrees of freedom
\[ \text{df} = n - 1\]
\[ = 40 - 1 \]
\[= 39\]
We can find the critical value using a t-distribution table or calculator. The critical value for a left-tailed test with $α = 0.01$ and $df = 39$ is approximately $-2.426$.
