Question

Red Blood Cell Count A simple random sample of 40 adult males is obtained, and the red blood cell count (in cells per microliter) is measured for each of them, with these results: n = 40, x = 4.932 million cells per microliter, s = 0.504 million cells per microliter (from Data Set 1 “Body Data” in Appendix B). Use a 0.01 significance level to test the claim that the sample is from a population with a mean less than 5.4 million cells per microliter, which is often used as the upper limit of the range of normal values. Does the result suggest that each of the 40 males has a red blood cell count below 5.4 million cells per microliter?

Short answer

The null hypothesis is rejected, which implies that there is enough evidence to support the claim that the mean blood cell count of the population of adult males is lesser than 5.4 million cells per microliter.

The result is valid for the distribution of sample means. It does not suggest anything about the individual counts of each sampled male.

Step-by-step solution

Given information

The results for the red blood cell count for 40 adult males in million cells per microliter are summarized below.

Sample size $n=40$.

Sample mean $\overline{x}=4.932$.

Sample standard deviation $s=0.504$.

Level of significance $\alpha =0.01$.

The claim states that the mean of the population is less than 5.4 million per microliter.

State the hypotheses

Null hypothesis: The sample is from the mean equal to 5.4 million cells per microliter.

Alternative hypothesis: The sample is from the mean below 5.4 million cells per microliter.

Mathematically,

$$\begin{gathered} H_0:\mu =5.4 \\ {H}_1:\mu <5.4 \end{gathered}$$

Here, $\mu$is the actual population mean red blood cell counts.

Compute the test statistic

Assume that the simple random sample is taken from a normally distributed population. For an unknown measure of population standard deviation, the t-test applies.

The test statistic is given as follows.

$$\begin{gathered} t=\frac{\overline{x}-\mu }{\frac{s}{\sqrt{n}}} \\ =\frac{4.932-5.4}{\frac{0.504}{\sqrt{40}}} \\ =-5.873 \end{gathered}$$

Compute the critical value

The degree of freedom is computed as follows.

$$\begin{gathered} df=n-1 \\ =40-1 \\ =39 \end{gathered}$$

Refer to the t-table for the critical value corresponding to the row with degrees of freedom 39 and the level of significance 0.01 for the one-tailed test.

The critical value is$t_{0.01}=-2.426$

State the decision

If the test statistic is lesser than the critical value, reject the null hypothesis. Otherwise, fail to reject the null hypothesis.

The critical value is not lesser than the test statistic, and thus, the null hypothesis is rejected at a 0.01 level of significance.

Thus, there is sufficient evidence to support the claim that the mean of red blood cell count is lesser than 5.4 million cells per microliter.

Analyze the result

The result suggests that the mean of the counts of cells for the population of adult males is lesser than 5.4 million per microliter. It does not infer anything about the count of cells for each male.