Question

Question 1

A pharmaceutical company carried out a study to investigate if men and women, of the same age, have different levels of the high density lipoprotein (HDL) type of cholesterol. This is the type of cholesterol that is labelled “good” as it helps prevent coronary disease. A small volunteer sample of men and women aged 40-45 of general good health was taken as a preliminary trial. Their blood HDL cholesterol levels (mg/dL) were measured with the following results:

MALE: 40.4 44.6 41.2 45.6 51.0 49.5

FEMALE: 52.2 49.9 61.0 53.5 59.5 48.0 55.0

a) Conduct a test of significance on the difference in mean HDL levels for men and women to test the hypothesis that the mean HDL is greater for females. Use a significance level of 0.05. Follow all steps clearly and write a clear conclusion. Show your graphical evidence, if needed.

b) What is the 95% confidence interval for the difference in mean HDL levels between all men and women? Interpret your interval in context.

Answer 1

Answer:-

Given That:-

A pharmaceutical company carried out a study to investigate if men and women, of the same age, have different levels of the high density lipoprotein (HDL) type of cholesterol. This is the type of cholesterol that is labelled “good” as it helps prevent coronary disease. A small volunteer sample of men and women aged 40-45 of general good health was taken as a preliminary trial.

a) Conduct a test of significance on the difference in mean HDL levels for men and women to test the hypothesis that the mean HDL is greater for females. Use a significance level of 0.05. Follow all steps clearly and write a clear conclusion. Show your graphical evidence, if needed.

Suppose randm variables X and Y denote blood HDL cholestrol level (in mg/dL) for female and male respectively.

Here, two different groups female and male are used to collect data. Further we do not know population standard deviation for variables. So, we have to perform two sample t - test.

We have to test for null hypothesis  

against the alternative hypothesis

Our test staistic is given by

Here,

First sample size

Second sample size

Sample mean of first sample is given by

= 54.15714

Sample mean of second sample is given by

= 45.38333

Pooled sample variance is given by

s = 4.551378

= 3.46496

Degrees of freedom

= 11

P - value =

= 0.002643363 [Using R - code 1 - pt(3.46496,11)]

Level of signifiance

We reject our null hypothesis if p - value <  

Here, we observe that p - value = 0.02643363 < 0.05 =

So we reject our null hypothesis

Hence, based on the given data we can conclude that there is significant that mean HDL level is greater for females.

b) What is the 95% confidence interval for the difference in mean HDL levels between all men and women? Interpret your interval in context.

Corresponding staistic is given by

Here,

First sample size

Second sample size

Sample mean of first sample is given by

= 54.15714

Sample mean of second sample is given by

= 45.38333

Pooled sample variance is given by

s = 4.551378

Degrees of freedom

= 11

We know,

[Using R-code qt(1-(1-0.95)/2,11)]

Interpretation:-

i) If we obtain such intervals using different samples, on average in 95% instances difference in population mean HDL levels between men and women is within our calculated intervals.

ii) Our calculated interval does not contain 0 (which corresponds to indifference between population means) So, with 95% confidence we can conclude that population mean HDL levels between all men and women differ.

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