5 stat questions complete solutions correct answers key

5 stat questions complete solutions correct answers key

1) We would like to estimate the true mean amount of time (in minutes) Canadian teenagers spend on the internet per day.  The population standard deviation of daily internet usage time is known to be 55 minutes.

(a)  What sample size is required in order to estimate  to within 15 minutes with 90% confidence?   

(b)  What sample size is required in order to estimate  to within 5 minutes with 90% confidence?  

(c)  For a fixed confidence level, when we decrease the desired margin of error to one third its original value, how many times larger a sample size will be require?

(d)  What sample size is required in order to estimate  to within 15 minutes with 98% confidence?  

(e)  For a fixed margin of error, when we increase the desired confidence level, what happens to the required sample size? (increase, decrease or remain the same).

(f)  The United States has ten times the population of Canada.  Assuming equal standard deviations, how many American teenagers must be sampled in order to estimate the true mean amount of time American teenagers spend on the internet per day to within 15 minutes with 90% confidence?

2)

An RCMP officer would like to estimate the true mean speed of all vehicles along a particular stretch of highway.  She measures the speeds of a simple random sample of 52 vehicles and calculates a mean speed of 112.16 km/h.  The population standard deviation of speeds is known to be 12.95 km/h.

(a)  What is a 97% confidence interval for the true mean speed of all vehicles on the highway?

(b) What is a 99% confidence interval for the true mean speed of all vehicles on this highway?

(c)  When the confidence level increases, What happens to the length of the confidence interval? Does it increase or decrease?

3)

A random variable X follows a normal distribution with standard deviation 13.  A random sample of 30 individuals is selected from the population, and a confidence interval for  is calculated to be (87.348, 96.652).  What is the confidence level for this interval?

4)

The Environmental Protection Agency (EPA) warns communities when their tap water is contaminated with too much lead.  Drinking water is considered unsafe if the mean concentration of lead is 15.1 parts per billion or greater.  The EPA would like to conduct a hypothesis test at the 10% level of significance to determine whether there is significant evidence that the tap water in one particular community is safe.  They randomly select 26 water samples from the community and calculate a mean lead concentration of 14.67 parts per billion.  Lead concentrations in the community are known to follow a normal distribution wtih standard deviation 2.41 parts per billion.  


(a)  What are the hypotheses for the appropriate test of significance?

(b) What is the value of the test statistic (to two decimal places)? 


(c)  What is the P-value of the test (to four decimal places)?


(d)  What is the appropriate conclusion for this test?

5)

A city's fire department would like to conduct a hypothesis test at the 1% level of significance to determine if their mean response time is greater than the target time of 7 minutes.  A random sample of 49 responses is timed, resulting in a mean of 8.4 minutes.  Response times are known to follow some right-skewed distribution with standard deviation 3.57 minutes.

(a)  Despite the fact that response times do not follow a normal distribution, it is still appropriate to use inference methods which rely on the assumption of normality.  This is because, of the Central Limit Theorem What does the Central Limit Theorem states?

(b)  What are the hypotheses for the appropriate test of significance?

(c)  What is the test statistic (to two decimal places)?


(d)  What is the P-value of the test (to four decimal places)? 


(e)  What is the correct conclusion of the test?