4. Are women getting taller? In one state, the average height of a woman aged 20 years or older is 63.7 inches in 1990. A random sample of 100 women is taken to test if women’s mean height today is different from 1990. The mean height of the 100 surveyed women is 63.9 inches. Assume that the sample’s standard deviation is 3.5 inches. Test the claim at the 0.05 level of significance.

a) State the null and alternative hypothesis
b) Estimate the standard error of the sampling distribution
c) Find the value of the test statistic and its corresponding p value
d) State your conclusion.

a) Ho: mean heights =

Ha: mean heights ≠

b) SEm = SD/√n

c) Z = (score-mean)/SEm

Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability related to the Z score.

d) Once you have done the calculations, you can come to your own conclusion.

a) The null hypothesis (H0) is that the mean height of women today is the same as in 1990. The alternative hypothesis (Ha) is that the mean height of women today is different from 1990.

b) The standard error of the sampling distribution can be estimated using the formula:

Standard Error = Standard Deviation / √Sample Size

In this case, the sample standard deviation is given as 3.5 inches, and the sample size is 100. Therefore,

Standard Error = 3.5 / √100 = 0.35 inches

c) To find the value of the test statistic and its corresponding p-value, we can use the Z-test. The formula for the test statistic is:

Z = (Sample Mean - Population Mean) / Standard Error

Here, the sample mean is 63.9 inches, and the population mean (from 1990) is 63.7 inches. The standard error is 0.35 inches. Plugging these values into the formula:

Z = (63.9 - 63.7) / 0.35

Calculating this gives us the value of the test statistic.

To find the corresponding p-value, we can then look up the test statistic in a standard normal distribution table or use a statistical calculator.

d) With the calculated test statistic and p-value, we can make a conclusion based on the level of significance (α) given as 0.05.

If the p-value is less than α (0.05), we reject the null hypothesis. This would indicate that there is evidence to suggest that women's mean height today is different from 1990.

If the p-value is greater than α (0.05), we fail to reject the null hypothesis. This would mean that there is not enough evidence to support the claim that women's mean height today is different from 1990.