Question

2. A teddy bear factory is found to be leaking a harmful substance called “luv”. As long as the average amount of luv in the ground soil is less than 4mg per meter, it will be safe. You are part of team determining whether the amount of luv is dangerous or not.

(a) [4 pts] Your team gathers 100 soil samples and finds that the average amount of luv is 3.5mg per meter, with a deviation of 0.7mg. Conduct an hypothesis test to see if there is too much luv. Remember, you need to give me H0, H1, and an appropriate p-value. You will be analyzing your answer in part (c).

(b) [4 pts] What is the smallest confidence interval that would capture the 4mg per meter safety limit?

(c) [2 pts] If your findings need to be at the 1% significance level, should the town worry? Explain your answer.

3. We are in the same situation as the previous problem. Only this time your team can only gather 3 samples. The average in the samples is still 3.5mg per meter, with a deviation of 0.7mg.

(a) [4 pts] Conduct an hypothesis test to see if there is too much luv. Remember, you need to give me H0, H1, and an appropriate p-value. You will be analyzing your answer in part (c).

(b) [4 pts] What is the smallest confidence interval that would capture the 4mg per meter safety limit?

(c) [2 pts] Using a 1% signficance level sShould the town worry? Explain your answer.

Answer 1

dear student, we can provide you with the solution of 1 question and 4 subquestion at . a time.

a) This is a one tailed t test for mean

since the population standard deviation is not mentioned we will use t test

null hypothesis :

alternate hypothesis: we will test if the luv is less than 4

Test statistic:

degree of freedom = n-1=100-1=99

The P value for the left -tailed test ( t distribution is symmetric)

b) The margin of error in the confidence interval is

the sample mean is 3.5

for the confidence interval to capture the limit 4 the upper limit of the confidence limit should be 4.

i.e the margin of error should be 0.5 at least

for 99 degrees of freedom

P value corresponding to this is 0.9999

Hence the smallest confidence interval is 9999%

c) at 0.01 significance level, we can reject the null hypothesis

as the P value is less than the significance level. we have evidence that the mean content of luv is less than 4mg.

Hence the town should not worry as luv is less than 4 mg

3) a) now the sample size is n= 3

null hypothesis :

alternate hypothesis: we will test if the luv is less than 4

Test statistic:

degree of freedom = n-1=3-1=2

The P value for the left -tailed test ( t distribution is symmetric) =0.1708

b) The smallest confidence interval that will contain the 4mg is (1-0.1708)=0.8292=82.92%

c) Since the P-value is greater than the significance level, we fail to reject the null hypothesis

Hence the people should worry. As the mean content of luv is not less than 4 mg