Question

The average height of men in 1960 was found to be 68 inches (5 feet, 8 inches). A researcher claims that men today are taller than they were in 1960 and would like to test this hypothesis at the 0.01 significance level. The researcher randomly selects 73 men and records their height to find an average of 69.8244 inches with standard deviation of 2.2354 inches. What is the value of the test statistic?

Answer 1

Answer:

t=0.477

$p_v =P(t_{(72)}>0.477)=0.317$

Step-by-step explanation:

1) Data given and notation

$\bar X=69.8244$ represent the mean height for the sample  

$s=2.2354$ represent the sample standard deviation for the sample

$n=73$ sample size

$\mu_o =68$ represent the value that we want to test

$\alpha=0.01$ represent the significance level for the hypothesis test.

t would represent the statistic (variable of interest)

$p_v$ represent the p value for the test (variable of interest)

Part a: State the null and alternative hypotheses.

We need to conduct a hypothesis in order to check if the mean height actually is higher than the mean height for men in 1960, the system of hypothesis would be:

Null hypothesis:$\mu \leq 68$

Alternative hypothesis:$\mu > 68$

If we analyze the size for the sample is > 30 but we don't know the population deviation so is better apply a t test to compare the actual mean to the reference value, and the statistic is given by:

$t=\frac{\bar X-\mu_o}{\frac{s}{\sqrt{n}}}$  (1)

t-test: "Is used to compare group means. Is one of the most common tests and is used to determine if the mean is (higher, less or not equal) to an specified value".

Part b: Calculate the statistic

We can replace in formula (1) the info given like this:

$t=\frac{69.8244-68}{\frac{2.2354}{\sqrt{73}}}=0.477$  

Part c: P-value

The first step is calculate the degrees of freedom, on this case:

$df=n-1=73-1=72$

Since is a one side test the p value would be:

$p_v =P(t_{(72)}>0.477)=0.317$

Part d: Conclusion

If we compare the p value and the significance level given $\alpha=0.01$ we see that $p_v>\alpha$ so we can conclude that we have enough evidence to fail reject the null hypothesis, so we can't conclude that the height of men actually it's significant higher compared to the height of men in 1960 at 1% of signficance.