Confidence Intervals & Hypothesis Testing
Confidence Intervals & Hypothesis Testing The National Patent Development Corporation (NPD) has recently acquired a new product that can be used to replace the dentist’s drill. The product, called Caridex, is a solution that dissolves decayed matter in cavities without requiring drilling. It is known that 100,000 dentists in the United States treat cavities. A preliminary analysis has revealed that 10% of all dentists would use Caridex in the first year after its introduction. However, Caridex is only effective on certain types of cavities. In a survey of 100 dentists, randomly selected from each state, each dentist was asked to estimate the number of cavities each week he or she would treat with Caridex. The results are given below. The dispensing unit costs NPD $200, and it intends to sell the unit at cost price. The solution costs $0.75 per cavity and will be sold to dentists at a price of $3.75 per cavity. Fixed annual costs are expected to be $2 million. NPD would like an estimate of the profit it can expect in the first year of operation.
Notes: There are 52 weeks in a year 7 4 3 4 2 7
3 2 5 3 6 4
1.
4 3 2 6 6 5
9 4 4 4 3 2
6 4 2 4 3 4
1 4 4 3 4 2
2 4 3 5 7 4
4 4 3 6 3 3
Profit = Revenue - Cost 5 2 3 6 3 3
3 6 7 5 4 3
6 7 6 5 3 7
4 1 3 7 3 6
4 2 3 5 7 3
6 4 2 1 2 3
4 5 6 5 5 1
3 6 2 5 2
4 6 5 3 2
(5 points) MINITAB Output. Enter the data in one column. Use Cavities as your header. Select Stat > Basic Statistics > 1 – sample t… The variable is Cavities. Click OK. MINITAB
will yield a 95% confidence interval for the mean number of cavities treated weekly by Caridex. 2.
(5 points) Although your sample is greater than 30, why is the t distribution appropriate
3.
(5 points) Using the interval from question 1, determine the 95% confidence interval
for problem? What assumption is necessary about the population?
estimate of the annual number of cavities to be treated by Caridex. Show your calculations (Hint: this number should be in the millions) 4.
(5 points) Using the interval from question 3, determine the 95% confidence interval
estimate of the annual profit. Show your calculations
One of the major U. S. producers of household products recently surveyed 64 adults in order to estimate proportion of adults who prefer mint-flavored toothpaste to plain toothpaste. The motive behind the survey was to aid in its production planning efforts. The results of the survey are listed below. Mint Plain Plain Mint Mint Plain Mint Mint
Mint Mint Mint Plain Mint Mint Mint
Mint Plain Plain Mint Plain Mint Mint
Plain Plain Plain Mint Mint Plain Mint
Plain Mint Mint Mint Mint Plain Mint
Mint Mint Mint Mint Mint Mint Plain
Mint Plain Mint Plain Plain Plain Mint
Plain Plain Mint Mint Mint Mint Mint
Mint Plain Mint Plain Plain Mint Plain
5.
(5 points) MINITAB Output. Enter these data into a MINTAB worksheet all in one column.
6.
(5 points) Based on the 99% confidence interval for adults who prefer plain toothpaste,
Use Toothpaste as your header. Select Stat > Basic Statistics > 1-Proportion… Complete the dialog in order to obtain a 99% confidence interval. do you have sufficient evidence to conclude that more than 60% of adults prefer mint toothpaste? Explain.
The owner of a restaurant that serves continental food wants to study characteristics of customers of her restaurant. In particular, she decides to focus on two variables: the amount of money spent by customers and whether customers order dessert. 7.
(5 points) If she wants to have 95% confidence of estimating the population mean
8.
(5 points) If she wants to have 90% confidence of estimating the population proportion
9.
(5 points) Based on your answers to (7) and (8), what size sample should be taken?
amount spent in her restaurant to within ± $1.50 and the standard deviation is assumed to $8, what sample size is needed? Show your calculations of customers who purchase dessert to within ± 0.04, what sample size is needed? Show your calculation Explain your reason.
The Cell Tone Company sells cellular phones and airtime in several southeast states. At a recent meeting, the marketing manager stated that the average age of its customers is under 40. This came up in conjunction with a proposed advertising plan that is to be directed toward a young audience. Before actually completing the advertising plan, Cell Tone decided to randomly sample customers. Among the questions asked in the survey of 15 customers in the Jacksonville, Florida, area was the customers’ age. The following data was obtained: 34 36
42
35
63
37
37
52
58
25
28
34
36
28
29
The marketing manager must support his statement concerning average customer age in an upcoming board meeting. 10.
(5 points) State the null hypothesis and the alternative hypothesis.
11.
(5 points) State the critical value rule using the 10 percent level of significance.
12.
(5 points) Calculate your test statistic.
13.
(5 points) What is your decision regarding the null statement?
14.
(5 points) Write a conclusion statement.
15.
(5 points) Consider the result of the hypothesis test you conducted above. Which of the
Show your calculation
two types of hypothesis test errors could you have committed? Explain the ramification of committing such an error in the context of this scenario.
The claim that “40% of those persons who retired from an industrial job before the age of 60 would return to work if a suitable job was available,” is to be investigated at the 0.02 level of significance. If 74 out of 200 workers sampled said they would return to work, what is our decision? 16.
(5 points) State the null and alternative hypotheses.
17.
(5 points) Calculate the test statistic. Show your calculation
18.
(5 points) Determine the p-value.
19.
(5 points) What is your decision regarding the null statement if = 0.02?
20.
(5 points) Write a conclusion statement.
Notes: There are 52 weeks in a year 7 4 3 4 2 7
3 2 5 3 6 4
1.
4 3 2 6 6 5
9 4 4 4 3 2
6 4 2 4 3 4
1 4 4 3 4 2
2 4 3 5 7 4
4 4 3 6 3 3
Profit = Revenue - Cost 5 2 3 6 3 3
3 6 7 5 4 3
6 7 6 5 3 7
4 1 3 7 3 6
4 2 3 5 7 3
6 4 2 1 2 3
4 5 6 5 5 1
3 6 2 5 2
4 6 5 3 2
(5 points) MINITAB Output. Enter the data in one column. Use Cavities as your header. Select Stat > Basic Statistics > 1 – sample t… The variable is Cavities. Click OK. MINITAB
will yield a 95% confidence interval for the mean number of cavities treated weekly by Caridex. 2.
(5 points) Although your sample is greater than 30, why is the t distribution appropriate
3.
(5 points) Using the interval from question 1, determine the 95% confidence interval
for problem? What assumption is necessary about the population?
estimate of the annual number of cavities to be treated by Caridex. Show your calculations (Hint: this number should be in the millions) 4.
(5 points) Using the interval from question 3, determine the 95% confidence interval
estimate of the annual profit. Show your calculations
One of the major U. S. producers of household products recently surveyed 64 adults in order to estimate proportion of adults who prefer mint-flavored toothpaste to plain toothpaste. The motive behind the survey was to aid in its production planning efforts. The results of the survey are listed below. Mint Plain Plain Mint Mint Plain Mint Mint
Mint Mint Mint Plain Mint Mint Mint
Mint Plain Plain Mint Plain Mint Mint
Plain Plain Plain Mint Mint Plain Mint
Plain Mint Mint Mint Mint Plain Mint
Mint Mint Mint Mint Mint Mint Plain
Mint Plain Mint Plain Plain Plain Mint
Plain Plain Mint Mint Mint Mint Mint
Mint Plain Mint Plain Plain Mint Plain
5.
(5 points) MINITAB Output. Enter these data into a MINTAB worksheet all in one column.
6.
(5 points) Based on the 99% confidence interval for adults who prefer plain toothpaste,
Use Toothpaste as your header. Select Stat > Basic Statistics > 1-Proportion… Complete the dialog in order to obtain a 99% confidence interval. do you have sufficient evidence to conclude that more than 60% of adults prefer mint toothpaste? Explain.
The owner of a restaurant that serves continental food wants to study characteristics of customers of her restaurant. In particular, she decides to focus on two variables: the amount of money spent by customers and whether customers order dessert. 7.
(5 points) If she wants to have 95% confidence of estimating the population mean
8.
(5 points) If she wants to have 90% confidence of estimating the population proportion
9.
(5 points) Based on your answers to (7) and (8), what size sample should be taken?
amount spent in her restaurant to within ± $1.50 and the standard deviation is assumed to $8, what sample size is needed? Show your calculations of customers who purchase dessert to within ± 0.04, what sample size is needed? Show your calculation Explain your reason.
The Cell Tone Company sells cellular phones and airtime in several southeast states. At a recent meeting, the marketing manager stated that the average age of its customers is under 40. This came up in conjunction with a proposed advertising plan that is to be directed toward a young audience. Before actually completing the advertising plan, Cell Tone decided to randomly sample customers. Among the questions asked in the survey of 15 customers in the Jacksonville, Florida, area was the customers’ age. The following data was obtained: 34 36
42
35
63
37
37
52
58
25
28
34
36
28
29
The marketing manager must support his statement concerning average customer age in an upcoming board meeting. 10.
(5 points) State the null hypothesis and the alternative hypothesis.
11.
(5 points) State the critical value rule using the 10 percent level of significance.
12.
(5 points) Calculate your test statistic.
13.
(5 points) What is your decision regarding the null statement?
14.
(5 points) Write a conclusion statement.
15.
(5 points) Consider the result of the hypothesis test you conducted above. Which of the
Show your calculation
two types of hypothesis test errors could you have committed? Explain the ramification of committing such an error in the context of this scenario.
The claim that “40% of those persons who retired from an industrial job before the age of 60 would return to work if a suitable job was available,” is to be investigated at the 0.02 level of significance. If 74 out of 200 workers sampled said they would return to work, what is our decision? 16.
(5 points) State the null and alternative hypotheses.
17.
(5 points) Calculate the test statistic. Show your calculation
18.
(5 points) Determine the p-value.
19.
(5 points) What is your decision regarding the null statement if = 0.02?
20.
(5 points) Write a conclusion statement.
