Hello, I need your help with these questions

Please with R studio

Thanks

1) A researcher would like to know if rats fed a high protein diet gain more weight than rats fed a low protein diet. She randomly selects ten rats to feed a high protein diet, and twelve to feed a low protein diet. Their weight gain in grams after 60 days on the diet is measured. The results are shown below:

High Protein Low Protein

n=10 n=12

mean=125 gramsmean=103 grams

Standard deviation=18g Standard deviation=21g

a)Find the standard error of the mean difference

b)Calculate the t-statistic for the mean difference, assuming that the true mean difference is 0.

c) Find the 95% confidence interval for the mean difference

2)

The following data gives the number of miles per gallon for a random sample of U.S. cars vs. Japanese cars. We would like to test the hypothesis that Japanese cars get better mileage than U.S. cars at an alpha level of 0.01.

US Japanese

18 24

15 27

16 27

16 25

17 31

15 30

14 24

16 19

16 28

15 23

a)State the null and alternative hypotheses

b)Check the assumptions and conditions. Are they satisfied? Give your histograms for the nearly normal condition (use 5 bins).

c)Determine the critical value .

d)Calculate the statistic.

e)State the result of the hypothesis test. What can you conclude?

3) Since the formula for the degrees of freedom is complicated, an approximation is sometimes used. The rule is to choose

df = min(n1 - 1 , n2 - 1)

Repeat exercise 1 using this rule. Compare the two confidence intervals you determined. Which is larger? Why?

4) Read in the RealEstate dataset. This gives a random sample of 1047 homes and includes the price (in dollars), living area (in sq. ft.), bathrooms (#), bedrooms (#), lot size (acres), age (years), and fireplace (1=yes, 2=no). A real estate agent receives a commission for each home they sell. To guide their housing choices to maximize their commission, the agent would like to examine whether homes with fireplaces do in fact sell for more than homes without.

Calculate the sample mean price of homes with and without fireplaces. Test whether the means of the two populations are equal at the 0.01 level. What do you conclude?

RealEstate dataset available here: https://drive.google.com/file/d/1xlMlmhV4fBi7PBdkwLK5HIzJzhA0KdUg/view?usp=sharing

5) With the same RealEstate dataset

The agent would also like to see if older homes sell for the same price as newer homes. Define and old home as one that is greater than 30 years old, and a new home as one that is less than or equal to 30 years old.

Test the hypothesis that old and new homes sell for the same price at the 0.05 level. Interpret your results.

make a bar plot of the old and new home means with bars showing the 95% confidence intervals.