Question

It is known that the average hemoglobin of normal adult men is 140g/L. This research randomly investigates 60 adult men in a factory. The mean hemoglobin is 125g/L and the standard deviation is 15g/L. A researcher considered that the average hemoglobin of adult men in this factory is lower than that of general adult men. (1) Is this conclusion correct? Why? (10 points) (2) If we want to estimate the average hemoglobin of adult men in this factory, what statistical analysis method can we use? What are the results? (20 points) (3) To compare the average hemoglobin of adult men in this factory with that of normal adult men, what method can we use? Write down the complete analysis steps and draw a conclusion. ,

Answer 1

The provided sample mean is $\bar{X}=125$ and the sample standard deviation is $s=15,$ and the sample size is $n=60.$

1) The following null and alternative hypotheses need to be tested:

$H_0:\mu\geq140$

$H_1:\mu<140$

This corresponds to a left-tailed test, for which a t-test for one mean, with unknown population standard deviation will be used.

The number of degrees of freedom are $df=60-1=59,$ and the significance level is  $\alpha=0.05.$ The critical value for a left-tailed test is $t_c=-1.671093.$

The rejection region for this left-tailed test is $R=\{t:t<-1.671093\}$

The t-statistic is computed as follows:

$$t=\dfrac{\bar{X}-\mu}{s/\sqrt{n}}=\dfrac{125-140}{15/\sqrt{60}}=-7.7460$$

Since it is observed that $t=-7.7460<-1.671093=t_c,$ it is then concluded that the null hypothesis is rejected.

Therefore, there is enough evidence to claim that the population mean $\mu$ is less than 140, at the 0.05 significance level.

Using the P-value approach: for the left-tailed test , 59 degrees of freedom and $t=-7.7460$ from the table we find that the p-value is $p<0.0001\approx0,$ and since $p=0<0.05=\alpha,$ it is concluded that the null hypothesis is rejected.

Therefore, there is enough evidence to claim that the population mean $\mu$ is less than 140, at the 0.05 significance level.

The conclusion is correct.

2) A t-test for one mean, with unknown population standard deviation will be used.

The number of degrees of freedom are $df=60-1=59,$ and the significance level is $\alpha=0.05.$The critical t-value is $t_c=2.001.$  

The 95% confidence for the population mean $\mu$ is computed using the following expression

$$CI=(\bar{X}-t_c\times\dfrac{s}{\sqrt{n}}, \bar{X}+t_c\times\dfrac{s}{\sqrt{n}})$$

$$=(125-2.001\times\dfrac{15}{\sqrt{60}}, 125+2.001\times\dfrac{15}{\sqrt{60}})$$

$$=(121.125, 128.875)$$

The 95% confidence interval is $121.125<\mu<128.875.$  

3) From the Internet (https://www.mayoclinic.org/tests-procedures/hemoglobin-test/about/pac-20385075) we know that the normal range for hemoglobin for men is 135 to 175 g/L.

From 1) we conclude that there is enough evidence to claim that the average hemoglobin of adult men in this factory is lower than that of general adult men at the 0.05 significance level.