STAT Question | Complete Solution

1.) Active consumers make up 13.6% of the market and spend an average of $ 16.23 per month on your product. Passive consumers make up 23.8% of the market and spend $ 9.85. The remaining consumers have average spending of $ 14.77. Find the average spending for all consumers.

2.) A survey of 613 representative people within your current area indicates that they plan to spend a total of $ 2,135 on your products next year. You are considering expansion to a new city with 2.1 million people.
a. Find the average expenditure per person based on the survey of your current area.
b. If you can gain the same market presence in the new city that you have in your current area, what annual sales level do you expect?
 c. If you expect to gain in the new city only 60% of the market presence in your current area, what annual sales level do you expect?

3.) Here are weights of recently produced sinks: 20.8, 20.9, 19.5, 20.8, 20.0, 19.8, 20.1, 20.5, 19.8, 20.3, 20.0, 19.7, 20.3, 19.5, 20.2, 20.2, 19.5, 20.5 Find the usual summary measure that tells approximately how far from average these weights are.

4.) Here are first-year sales (in thousands) for some recent new product introductions that are similar to one you are considering.
10, 12, 16, 47, 39, 22, 10, 29
 a. Find the average and standard deviation. Interpret the standard deviation.
b. After you went ahead with the product introduction you were considering, it turned out to have first-year sales of 38 (thousand). How far from average is this? Compare this answer to the standard deviation.
 c. The next new product introduction rang up 92 (thousand) the first year. How many standard deviations (from part a) is this from the average (also from part a)?
d. For the new product introductions of parts b and c, say whether each is typical for your firm and indicate how you know.

5.) You are operations manager for a plant that produces copy machines. At the end of the day tomorrow, you will find out how many machines were produced and, of these, how many are defective. a. Describe the random experiment identified here.
b. What is the sample space? (Hint: Note that each outcome consists of two pieces of information.)
c. What will the outcome tell you?
d. Identify precisely the event “met the goal of at least 500 working (nondefective) machines with 2 or fewer defective machines” by listing the qualifying outcomes from the sample space.
 e. In 22 of the past 25 days, this goal has been met. Find the appropriate relative frequency.
f. About how far away from the true, unknown probability of meeting the goal is the relative frequency from part e? (For simplicity, you may assume that these were independent runs of the same random experiment.)

6.) Your company maintains a database with information on your customers, and you are interested in analyzing patterns observed over the past quarter. In particular, 23% of customers in the database placed new orders within this period. However, for those customers who had a salesperson assigned to them, the new order rate was 58%. Overall, 14% of customers within the database had salespeople assigned to them.
a. Draw a probability tree for this situation.
b. What percentage of customers in the database placed a new order but did not have a salesperson assigned to them?
 c. Given that a customer did not place a new order, what is the probability that the customer had a salesperson assigned to him or her?
d. If a customer did not have a salesperson assigned to him or her, what is the probability that the customer placed a new order?

7.) Your company is planning to market a new reading lamp and has segmented the market into three groups— avid readers, regular readers, and occasional readers— and currently assumes that 25% of avid readers, 15% of regular readers, and 10% of occasional readers will want to buy the new product. As part of a marketing survey, 400 individuals will be randomly selected from the population of regular readers. Using the current assumptions, find the mean and standard deviation of the percentage among those surveyed who will want to buy the new product.

8.) Suppose that the number of hits on your company's website, from noon to 1 p.m. on a typical weekday, follows a normal distribution (approximately) with a mean of 190 and a standard deviation of 24.
a. Find the probability that the number of hits is more than 160.
b. Find the probability that the number of hits is less than 215.
c. Find the probability that the number of hits is between 165 and 195.
d. Find the probability that the number of hits is not between 150 and 225.

9.) You have eight machines operating independently. The mean production rate for each machine is 20.3 tons per day, and the standard deviation is 1.4 tons per day. Approximately how much uncertainty is there in the average daily production for the eight machines? Please report the usual summary measure.

10.) Deposits have a mean of $ 125 and a standard deviation of $ 36. Find the standard deviation of the average amount of 12 randomly selected deposits.

11.) A survey of 21 business intelligence analysts, who had been in their current positions from 10 to 20 years, revealed $ 90,734 as the average salary.  Assume a random sample with a standard deviation of $ 15,000. Computerworld's Smart Salary Tool 2010, accessed at http:// www.computerworld.com/ s/ salary-survey/ tool/ 2010/ on July 13, 2010.
a. Find the 95% confidence interval for the population mean salary.
b. Find the 99% confidence interval for the population mean salary.
c. Complete the following sentence: We are 95% sure that the population mean salary is at least ______________________.

12. A survey of eight vice presidents of information technology and information systems, who had been in their current positions for 10 years or less, revealed $ 174,813 as the average salary.  Assume a random sample with a standard deviation of $ 25,000.
a. Find the 95% confidence interval for the population mean salary.
b. Find the 99% confidence interval for the population mean salary.
c. Complete the following sentence: We are 95% sure that the population mean salary is at least ______________________.

13.) Test whether or not the population percentage could reasonably be 20%, based on the observed 18.4% who like your products, from a random sample of 500 consumers.
14.) Managers perceived employee stock ownership as having a significant positive effect on product quality. As part of that same study, managers were also asked to rate the effect of employee stock ownership on unit labor cost. This effect, on a scale from –2 (large negative effect) to 2 (large positive effect), was 0.12 with a standard error of 0.11, based on a sample of 343 managers.
a. Find the 95% confidence interval and state carefully what this represents. Keep in mind that these are opinions of randomly selected managers.
 b. Is there a significant relationship between employee stock ownership and the unit cost of labor as perceived by managers? Why or why not?
c. Identify the null and research hypotheses.
d. Which hypothesis has been accepted? Is this a weak or a strong conclusion?
e. Has the accepted hypothesis been absolutely proven? If not, what type of error may have been made?