# Business Statistics with 95% confidence intervals

1. Determine the minimum required sample size if you want to be 95% confident that the sample mean is within 3 units of the population mean given sigma = 7.8. Assume the population is normally distributed.

2. A manufacturer claims that the mean lifetime of its fluorescent bulbs is 1050 hours. A homeowner selects 40 bulbs and finds the mean lifetime to be 1030 hours with a standard deviation of 80 hours. Test the manufacturer's claim. Use alpha = .04.

3. A marketing firm wants to estimate the average amount spent by shoppers leaving a mall. For a sample of 60 randomly selected shoppers, the mean amount spent was $54.20 and the standard deviation was $8.45.

(a) Find a 95% confidence interval for the mean amount spent by shoppers at this mall. Show your calculations.

(b) Interpret this confidence interval and write a sentence that explains it.

4. If the variance of some data is 225 inches², what is the standard deviation?

5. The Canadian Services Office did a survey of 150 travelers in which they asked if the traveler's first language was French or English. Another question asked was whether the traveler was Canadian born. The results follow.

Canadian Not Canadian Row Totals

English 15 50 65

French 65 20 85

Column Totals 80 70 150

If a traveler is selected at random (from this group of 150 travelers), find the probability that

P (The traveler speaks French)

A) 85/150

B) 65/85

C) 20/85

D) 18/30

6. Which of these is true of the correlation coefficient?

A) both 1 and 3

B) its range is [-1, 1]

C) we use r to represent it

D) we use r² to represent it

E) both 1 and 2

7. In a study done by the National Health Foundation, it was determined that the mean number of headaches for women is 14 per year with a standard deviation of 2.5. How many standard deviations is 16.50 from the mean?

A) 1.25

B) 0.75

C) 0.50

D) 1.00

E) 0.25

8. A social service worker wants to estimate the true proportion of pregnant teenagers who miss at least one day of school per week on average. The social worker wants to be within 4% of the true proportion when using a 95% confidence interval. A previous study estimated the population proportion at 0.28.

(a) Using this previous study as an estimate for p, what sample size should be used?

(b) If the previous study was not available, what estimate for p should be used?

9. The stem and leaf plot for the following data is displayed below:

{70, 78, 76, 55, 43, 56, 32, 67, 68, 71, 75, 67, 60, 62, 58, 75, 21}

Stem and Leaf Plot:

2|1

3|2

4|3

5|568

6|78702

7|086155

Discuss the shape of the data distribution.

10. Suppose you are performing a hypothesis test on a claim about a population proportion. Using an alpha = .05 and n = 100, what is the rejection region if the alternate hypothesis is Ha: p > 0.75?

A) Reject Ho if z < -1.645

B) Reject Ho if z > 1.96

C) Reject Ho if z > 1.645

D) Reject Ho if z > 2.33

11. The Canadian Services Office did a survey of 150 travelers in which they asked if the traveler's first language was French or English. Another question asked was whether the traveler was Canadian born. The results follow.

Canadian Not Canadian Row Totals

English 15 50 65

French 65 20 85

Column Totals 80 70 150

If a traveler is selected at random (from this group of 150 travelers), find the probability that

P(The traveler is not a Canadian and speaks French as the 1st language)

A) 50/150

B) 65/150

C) 15/80

D) 20/150

12. The Canadian Services Office did a survey of 150 travelers in which they asked if the traveler's first language was French or English. Another question asked was whether the traveler was Canadian born. The results follow.

Canadian Not Canadian Row Totals

English 15 50 65

French 65 20 85

Column Totals 80 70 150

If a traveler is selected at random (from this group of 150 travelers), find the probability that

P(The traveler is Canadian, given that the first language is English)

A) 65/80

B) 15/150

C) 80/85

D) 15/65

13. The random variable X represents the annual salaries in dollars of a group of teachers. Given the following probabilities: P(30,000) = .5; P(40,000) = .3; P(50,000) = .2, find the expected value E(X).

X = {$30,000; $40,000; $50,000}.

14. A car towing company averages 3 calls per hour. What is the probability that in a randomly selected hour the number of calls is 2?

15. A restaurant claims that its speed of service time is less than 18 minutes. A random selection of 36 service times was collected, and their mean was calculated to be 17.1 minutes. Their standard deviation is 3.1 minutes. Is there enough evidence to support the claim at alpha = .08. Perform an appropriate hypothesis test, showing each important step. (Note: 1st Step: Write Ho and Ha; 2nd Step: Determine Rejection Region; etc.)

16. Suppose you are performing a hypothesis test on a claim about a population proportion. Using an alpha = .05 and n = 90, what two critical values determine the rejection region if the null hypothesis is: Ho: p = 0.35?

A) ± 1.96

B) none of these

C) ± 1.28

D) ± 2.33

17. Find the following probability involving the Standard Normal Distribution. What is P(z<2.25)?

A) .9798

B) .9878

C) .9938

D) .0122

18. The heights of 8 4th graders are listed in inches. {50, 54, 55, 52, 59, 54, 54, 53}.

Find the mean, median, mode, variance, and range.

Do you think this sample might have come from a normal population? Why or why not?

19. The probability that a house in an urban area will be burglarized is 3%. If 45 houses are randomly selected what is the probability that one of the houses will be burglarized?

a. Is this a binomial experiment? Explain how you know.

b. Use the correct formula to find the probability that, out of 45 houses, exactly 3 of the houses will be burglarized. Show your calculations or explain how you found the probability.

20. The average (mean) monthly gasoline (gallon) purchase for a family with 2 cars is 62. This statistic has a normal distribution with a standard deviation = 6 gallons. A family is chosen at random.

a) Find the probability that the family's monthly gasoline (gallon) purchases will be between 47 and 67 gallons.

b) Find the probability that the family's monthly gasoline (gallon) purchases will be less than 67 gallons.

c) What is the probability that the family's monthly gasoline (gallon) purchases will be more than 47 gallons?

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#### Solution Summary

This solution is comprised of detailed step-by-step calculations and analysis of the given problems related to Statistics and provides students with a clear perspective of the underlying concepts.

Business Statistics: Relationship between Confidence Intervals & Hypothesis Testing

What is the relationship between confidence intervals and hypothesis testing? How are they the same? How are they different?

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