Solve what is asked in each of the points. It is important that you show your entire procedure to reach the result.
1. Mattel Corporation produces remote-controlled cars that run on AA batteries. The mean battery life for this product is 35.0 hours. The distribution of battery lives approximates a normal probability distribution with a standard deviation of 5.5 hours. As part of its program, Sony tests samples of 25 batteries.
a) What can be said about the shape of the sampling distribution of the mean?
b) What is the standard error of the sampling distribution of the mean?
c) What proportion of the samples will have a mean shelf life of more than 36 hours?
d) What proportion of the sample will have a mean shelf life greater than 34.5 hours?
e) What proportion of the sample will have a mean shelf life between 34.5 and 36 hours?
2. The mean amount of groceries purchased by each customer at Churchill Grocery Store is $23.50, with a standard deviation of $5.00. Assume that the distribution of quantities purchased follows the normal distribution. For a sample of 50 customers, answer the following questions.
a) What is the probability that the sample mean is at least $25.00?
b) What is the probability that the sample mean is greater than $22.50 and less than $25.00?
c) Within what limits will 90% of the sample means be presented?
3. Over the past decade, the average number of Information Systems Security Association members who experienced denial-of-service attacks each year was 510, with a standard deviation of 14.28 attacks. Assume nothing changes in this environment.
a) What is the probability that this group will suffer an average of more than 600 attacks over the next 10 years?
b) Calculate the probability that they will experience an average of 500 to 600 attacks over the next 10 years.
c) What is the probability that they will experience an average of fewer than 500 attacks over the next 10 years?
4. Marty Rowatti has just taken over as director of the YMCA of South Jersey. He would like recent data on how long his current members have been members of the YMCA. To investigate this, suppose he selects a random sample of 40 current members. The mean membership length of those in the sample is 8.32 years, and the standard deviation is 3.07 years.
a) What is the population mean?
b) Construct a 90% confidence interval for the population mean.
c) The previous director, in her brief report on her retirement, indicated that the average membership period was now “almost 10 years.” Does the information confirm this assertion? Please provide evidence.
5. A recent study by the American Automobile Dealers Association found that the mean amount of profit per car sold in a sample of 20 dealerships was $290, with a standard deviation of $125. Construct a 95% confidence interval for the population mean.
6. A survey of 36 randomly selected iPhone owners showed that the purchase price has a mean of $416, with a sample standard deviation of $180.
a) Calculate the standard error of the sample mean.
b) Calculate the 95% confidence interval of the mean.
c) What size should the sample be to estimate the population mean within $10?