The following table shows the summary statistic of the cholesterol levels of two groups of men in 2014.

From 25 to 40 years old, the mean is 224, standard deviation is 5.5 and the sample size is 12.
From 41 to 55 years old, the mean is 230, standard deviation is 7 and the sample size is 10.
It is assumed that the population standard deviations of cholesterol levels in both groups are the same.

a) In 2013, the mean cholesterol level of men from 25 to 40 years old is 218. Test the claim that
the mean cholesterol level of men from 25 to 40 years old has increased in 2014. Use the
P-value method. State the assumptions required.
b) A researcher claims that there is a difference in the cholesterol levels of men from 25 to 40 years
old and from 41 to 55 years old. From the data collected in 2014 above, is there enough evidence to support the claim? Conduct a two-sample test at the 5% significance level to explain your answer.

a) To test the claim that the mean cholesterol level of men from 25 to 40 years old has increased in 2014, we can use the P-value method. The null hypothesis (H0) is that the mean cholesterol level in 2014 is the same as 2013, while the alternative hypothesis (Ha) is that the mean cholesterol level in 2014 is greater than in 2013.

Assumptions:
1. The samples are independent.
2. The population standard deviation is the same as in 2013.

We can perform a one-sample Z-test to calculate the P-value. The formula for the test statistic (Z) is given by:
Z = (sample mean - hypothesized mean) / (standard deviation / √sample size)

Given:
Sample mean (2014) = 224
Hypothesized mean (2013) = 218
Standard deviation = 5.5
Sample size = 12

Calculating the test statistic (Z):
Z = (224 - 218) / (5.5 / √12)
Z = 6 / (5.5 / 3.464)
Z ≈ 6 / 1.588
Z ≈ 3.77

Next, we need to find the P-value associated with this Z-score using a standard normal distribution table or a statistical software. The P-value is the probability of observing a test statistic as extreme as the one we calculated, assuming the null hypothesis is true.

For a one-sided test with the alternative hypothesis of the mean being greater, we look up the Z-score in the right-tail of the standard normal distribution. The P-value is the probability of getting a Z-score greater than 3.77.

Based on the P-value, we can make a decision. If the P-value is less than the significance level (α), we reject the null hypothesis. If the P-value is greater than α, we fail to reject the null hypothesis.

b) To test the claim that there is a difference in the cholesterol levels between the two age groups (25-40 and 41-55), we can conduct a two-sample t-test. The null hypothesis (H0) is that the means of the two groups are equal, while the alternative hypothesis (Ha) is that the means are different.

Assumptions:
1. The samples are independent.
2. The populations have the same standard deviation.

We can calculate the test statistic (t) using the following formula:
t = (x1 - x2) / √[(s1^2 / n1) + (s2^2 / n2)]

Given:
Group 1 (25-40):
Mean (x1) = 224
Standard deviation (s1) = 5.5
Sample size (n1) = 12

Group 2 (41-55):
Mean (x2) = 230
Standard deviation (s2) = 7
Sample size (n2) = 10

Calculating the test statistic (t):
t = (224 - 230) / √[(5.5^2 / 12) + (7^2 / 10)]
t = -6 / √[(30.25 / 12) + (49 / 10)]
t = -6 / √(2.52 + 4.9)
t = -6 / √7.42
t ≈ -6 / 2.723
t ≈ -2.203

To determine if there is enough evidence to support the claim of a difference in cholesterol levels between the two age groups, we need to find the critical value associated with the chosen significance level (α) (in this case, 5%).

We compare the calculated t-value to the critical value. If the calculated t-value is greater than the critical value (or falls within the rejection region), we reject the null hypothesis. If the calculated t-value is less than the critical value (or falls within the acceptance region), we fail to reject the null hypothesis.

Alternatively, we can calculate the P-value associated with the t-value using a t-distribution table or statistical software. The P-value is the probability of observing a test statistic as extreme as the one we calculated, assuming the null hypothesis is true.

Using either the critical value or the P-value, we can make a decision about whether there is enough evidence to support the claim of a difference in cholesterol levels between the two age groups.