7.2hw

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Tutorial Exercise

Use the Student'stdistribution to findt_cfor a 0.95 confidence level when the sample is28.

Step 1

Recall that a Student'stdistribution table lists critical values

t_c

for acconfidence level. The table is arranged by column headings forcand by row headings for degrees of freedomd.f. Decide which column to use by identifying the confidence level value.c=

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The method of tree ring dating gave the following years A.D. for an archaeological excavation site. Assume that the population ofxvalues has an approximately normal distribution.

1,187

1,243

1,285

1,180

1,268

1,316

1,275

1,317

1,275

(a) Use a calculator with mean and standard deviation keys to find the sample mean yearxand sample standard deviations. (Round your answers to four decimal places.)

x=	A.D.
s=	yr

(b) Find a 90% confidence interval for the mean of all tree ring dates from this archaeological site. (Round your answers to the nearest whole number.)

lower limit	A.D.
upper limit	A.D.

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The method of tree-ring dating gave the following years A.D. for an archaeological excavation site. Assume that the population ofxvalues has an approximately normal distribution.

1,250

1,208

1,229

1,264

1,268

1,316

1,275

1,317

1,275

(a)

Use a calculator with mean and standard deviation keys to find the sample mean yearxand sample standard deviations. (Round your answers to four decimal places.)

x

= A.D.s= yr

(b)

When finding an 90% confidence interval, what is the critical value for confidence level? (Give your answer to three decimal places.)

t_c=

What is the maximal margin of error when finding a 90% confidence interval for the mean of all tree-ring dates from this archaeological site? (Round your answer to the nearest whole number.)

E=

Find a 90% confidence interval for the mean of all tree-ring dates from this archaeological site. (Round your answers to the nearest whole number.)

lower limit A.D.upper limit A.D.

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The method of tree-ring dating gave the following years A.D. for an archaeological excavation site. Assume that the population ofxvalues has an approximately normal distribution.

1,222

1,306

1,229

1,236

1,268

1,316

1,275

1,317

1,275

(a)

Use a calculator with mean and standard deviation keys to find the sample mean yearxand sample standard deviations. (Round your answers to four decimal places.)

x

= A.D.s= yr

(b)

Find a 90% confidence interval for the mean of all tree-ring dates from this archaeological site. (Round your answers to the nearest whole number.)

lower limit A.D.upper limit A.D.

5.

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How much does a sleeping bag cost? Let's say you want a sleeping bag that should keep you warm in temperatures from20Fto45F.A random sample of prices ($) for sleeping bags in this temperature range is given below. Assume that the population ofxvalues has an approximately normal distribution.

70	60	100	50	40	105	30	23	100	110
105	95	105	60	110	120	95	90	60	70

(a)

Use a calculator with mean and sample standard deviation keys to find the sample mean pricexand sample standard deviations. (Round your answers to four decimal places.)

x=$ s=$

(b)

Using the given data as representative of the population of prices of all summer sleeping bags, find a 90% confidence interval for the mean priceof all summer sleeping bags. (Round your answers to two decimal places.)

lower limit$ upper limit$

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How much do wild mountain lions weigh? Adult wild mountain lions (18 months or older) captured and released for the first time in the San Andres Mountains gave the following weights (pounds):

67

100

127

126

60

64

Assume that the population ofxvalues has an approximately normal distribution.(a) Use a calculator with mean and sample standard deviation keys to find the sample mean weightxand sample standard deviations. (Round your answers to four decimal places.)

x=	lb
s=	lb

(b) Find a 75% confidence interval for the population average weightof all adult mountain lions in the specified region. (Round your answers to one decimal place.)

lower limit	lb
upper limit	lb

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Do you want to own your own candy store? Wow! With some interest in running your own business and a decent credit rating, you can probably get a bank loan on startup costs for franchises such as Candy Express, The Fudge Company, Karmel Corn, and Rocky Mountain Chocolate Factory. Startup costs (in thousands of dollars) for a random sample of candy stores are given below. Assume that the population ofxvalues has an approximately normal distribution.

90

176

130

97

75

94

116

100

85

(a) Use a calculator with mean and sample standard deviation keys to find the sample mean startup costxand sample standard deviations. (Round your answers to four decimal places.)

x=	thousand dollars
s=	thousand dollars

(b) Find a 90% confidence interval for the population average startup costsfor candy store franchises. (Round your answers to one decimal place.)

lower limit	thousand dollars
upper limit	thousand dollars

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Over the past several months, an adult patient has been treated for tetany (severe muscle spasms). This condition is associated with an average total calcium level below 6 mg/dl. Recently, the patient's total calcium tests gave the following readings (in mg/dl). Assume that the population ofxvalues has an approximately normal distribution.

10.10

9.20

10.90

8.90

9.40

9.80

10.00

9.90

11.20

12.10

(a)

Use a calculator with mean and sample standard deviation keys to find the sample mean readingxand the sample standard deviations. (Round your answers to four decimal places.)

x= mg/dls= mg/dl

(b)

Find a 99.9% confidence interval for the population mean of total calcium in this patient's blood. (Round your answer to two decimal places.)

lower limit mg/dlupper limit mg/dl

(c)

Based on your results in part (b), do you think this patient still has a calcium deficiency? Explain.

Yes. This confidence interval suggests that the patient may still have a calcium deficiency.Yes. This confidence interval suggests that the patient no longer has a calcium deficiency. No. This confidence interval suggests that the patient may still have a calcium deficiency.No. This confidence interval suggests that the patient no longer has a calcium deficiency.

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What percentage of hospitals provide at least some charity care? Based on a random sample of hospital reports from eastern states, the following information is obtained (units in percentage of hospitals providing at least some charity care):

56.5

56.3

53.3

65.5

59.0

64.7

70.1

64.7

53.5

78.2

Assume that the population ofxvalues has an approximately normal distribution.

(a)

Use a calculator with mean and sample standard deviation keys to find the sample mean percentagexand the sample standard deviations. (Round your answers to four decimal places.)

x= %s= %

(b)

Find a 90% confidence interval for the population averageof the percentage of hospitals providing at least some charity care. (Round your answers to one decimal place.)

lower limit %upper limit %

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For this problem, carry at least four digits after the decimal in your calculations. Answers may vary slightly due to rounding. In a random sample of70professional actors, it was found that38were extroverts.

(a) Letprepresent the proportion of all actors who are extroverts. Find a point estimate forp. (Round your answer to four decimal places.) (b) Find a 95% confidence interval forp. (Round your answers to two decimal places.)

lower limit
upper limit

Give a brief interpretation of the meaning of the confidence interval you have found.We are 95% confident that the true proportion of actors who are extroverts falls within this interval.We are 95% confident that the true proportion of actors who are extroverts falls outside this interval. We are 5% confident that the true proportion of actors who are extroverts falls within this interval.We are 5% confident that the true proportion of actors who are extroverts falls above this interval.

(c) Do you think the conditionsnp>5 andnq>5 are satisfied in this problem? Explain why this would be an important consideration.No, the conditions are not satisfied. This is important because it allows us to say thatpis approximately binomial.No, the conditions are not satisfied. This is important because it allows us to say thatpis approximately normal. Yes, the conditions are satisfied. This is important because it allows us to say thatpis approximately binomial.Yes, the conditions are satisfied. This is important because it allows us to say thatpis approximately normal.

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For this problem, carry at least four digits after the decimal in your calculations. Answers may vary slightly due to rounding.

In a random sample of65professional actors, it was found that42were extroverts.

(a)

Letprepresent the proportion of all actors who are extroverts. Find point estimates forpandq. (Round your answer to four decimal places.)

p=q=

(b)

Find a 95% confidence interval forp.

Find the maximal margin of error. (Round your answer to two decimal places.)

E=

Report the bounds from the 95% confidence interval forp. (Round your answers to two decimal places.)

lower limitupper limit

Give a brief interpretation of the meaning of the confidence interval you have found.

We are 5% confident that the true proportion of actors who are extroverts falls within this interval.We are 95% confident that the true proportion of actors who are extroverts falls outside this interval. We are 5% confident that the true proportion of actors who are extroverts falls above this interval.We are 95% confident that the true proportion of actors who are extroverts falls within this interval.

(c)

Do you think the conditionsnp>5 andnq>5 are satisfied in this problem? Explain why this would be an important consideration.

No, the conditions are not satisfied. This is important because it allows us to say thatpis approximately normal.No, the conditions are not satisfied. This is important because it allows us to say thatpis approximately binomial. Yes, the conditions are satisfied. This is important because it allows us to say thatpis approximately binomial.Yes, the conditions are satisfied. This is important because it allows us to say thatpis approximately normal.

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[-/0.2 Points]DETAILSBBUNDERSTAT12 7.3.011.FB.S.

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For this problem, carry at least four digits after the decimal in your calculations. Answers may vary slightly due to rounding.

In a random sample of60professional actors, it was found that43were extroverts.

(a)

Letprepresent the proportion of all actors who are extroverts. Find a point estimate forp. (Round your answer to four decimal places.)

(b)

Find a 95% confidence interval forp. (Round your answers to two decimal places.)

lower limitupper limit

Give a brief interpretation of the meaning of the confidence interval you have found.

We are 5% confident that the true proportion of actors who are extroverts falls within this interval.We are 95% confident that the true proportion of actors who are extroverts falls within this interval. We are 95% confident that the true proportion of actors who are extroverts falls outside this interval.We are 5% confident that the true proportion of actors who are extroverts falls above this interval.

(c)

Do you think the conditionsnp>5 andnq>5 are satisfied in this problem? Explain why this would be an important consideration.

No, the conditions are not satisfied. This is important because it allows us to say thatpis approximately normal.No, the conditions are not satisfied. This is important because it allows us to say thatpis approximately binomial. Yes, the conditions are satisfied. This is important because it allows us to say thatpis approximately normal.Yes, the conditions are satisfied. This is important because it allows us to say thatpis approximately binomial.

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For this problem, carry at least four digits after the decimal in your calculations. Answers may vary slightly due to rounding.

In a random sample of530judges, it was found that285were introverts.

(a)

Letprepresent the proportion of all judges who are introverts. Find a point estimate forp. (Round your answer to four decimal places.)

(b)

Find a 99% confidence interval forp. (Round your answers to two decimal places.)

lower limitupper limit

Give a brief interpretation of the meaning of the confidence interval you have found.

We are 99% confident that the true proportion of judges who are introverts falls within this interval.We are 1% confident that the true proportion of judges who are introverts falls above this interval. We are 99% confident that the true proportion of judges who are introverts falls outside this interval.We are 1% confident that the true proportion of judges who are introverts falls within this interval.

(c)

Do you think the conditionsnp>5 andnq>5 are satisfied in this problem? Explain why this would be an important consideration.

No, the conditions are not satisfied. This is important because it allows us to say thatpis approximately normal.Yes, the conditions are satisfied. This is important because it allows us to say thatpis approximately normal. No, the conditions are not satisfied. This is important because it allows us to say thatpis approximately binomial.Yes, the conditions are satisfied. This is important because it allows us to say thatpis approximately binomial.

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For this problem, carry at least four digits after the decimal in your calculations. Answers may vary slightly due to rounding. A random sample of5060permanent dwellings on an entire reservation showed that1634were traditional hogans.

(a) Letpbe the proportion of all permanent dwellings on the entire reservation that are traditional hogans. Find a point estimate forp. (Round your answer to four decimal places.) (b) Find a 99% confidence interval forp. (Round your answers to three decimal places.)

lower limit
upper limit

Give a brief interpretation of the confidence interval.1% of all confidence intervals would include the true proportion of traditional hogans.99% of the confidence intervals created using this method would include the true proportion of traditional hogans. 99% of all confidence intervals would include the true proportion of traditional hogans.1% of the confidence intervals created using this method would include the true proportion of traditional hogans.

(c) Do you think thatnp>5 andnq>5 are satisfied for this problem? Explain why this would be an important consideration.No, the conditions are not satisfied. This is important because it allows us to say thatpis approximately normal.Yes, the conditions are satisfied. This is important because it allows us to say thatpis approximately binomial. No, the conditions are not satisfied. This is important because it allows us to say thatpis approximately binomial.Yes, the conditions are satisfied. This is important because it allows us to say thatpis approximately normal.

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For this problem, carry at least four digits after the decimal in your calculations. Answers may vary slightly due to rounding.

A random sample of5,120permanent dwellings on an entire reservation showed that1,662were traditional hogans.

(a)

Letpbe the proportion of all permanent dwellings on the entire reservation that are traditional hogans. Find a point estimate forp. (Round your answer to four decimal places.)

(b)

Find a 99% confidence interval forp. (Round your answers to three decimal places.)

lower limitupper limit

Give a brief interpretation of the confidence interval.

1% of all confidence intervals would include the true proportion of traditional hogans.99% of all confidence intervals would include the true proportion of traditional hogans. 1% of the confidence intervals created using this method would include the true proportion of traditional hogans.99% of the confidence intervals created using this method would include the true proportion of traditional hogans.

(c)

Do you think thatnp>5 andnq>5 are satisfied for this problem? Explain why this would be an important consideration.

Yes, the conditions are satisfied. This is important because it allows us to say thatpis approximately binomial.Yes, the conditions are satisfied. This is important because it allows us to say thatpis approximately normal. No, the conditions are not satisfied. This is important because it allows us to say thatpis approximately binomial.No, the conditions are not satisfied. This is important because it allows us to say thatpis approximately normal.

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For this problem, carry at least four digits after the decimal in your calculations. Answers may vary slightly due to rounding. Santa Fe black-on-white is a type of pottery commonly found at archaeological excavations at a certain monument. At one excavation site a sample of580potsherds was found, of which368were identified as Santa Fe black-on-white.

(a) Letprepresent the proportion of Santa Fe black-on-white potsherds at the excavation site. Find a point estimate forp. (Round your answer to four decimal places.) (b) Find a 95% confidence interval forp. (Round your answers to three decimal places.)

lower limit
upper limit

Give a brief statement of the meaning of the confidence interval.95% of all confidence intervals would include the true proportion of potsherds.5% of all confidence intervals would include the true proportion of potsherds. 95% of the confidence intervals created using this method would include the true proportion of potsherds.5% of the confidence intervals created using this method would include the true proportion of potsherds.

(c) Do you think thatnp>5 andnq>5 are satisfied for this problem? Explain why this would be an important consideration.No, the conditions are not satisfied. This is important because it allows us to say thatpis approximately normal.No, the conditions are not satisfied. This is important because it allows us to say thatpis approximately binomial. Yes, the conditions are satisfied. This is important because it allows us to say thatpis approximately normal.Yes, the conditions are satisfied. This is important because it allows us to say thatpis approximately binomial.

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For this problem, carry at least four digits after the decimal in your calculations. Answers may vary slightly due to rounding. Case studies showed that out of10,086convicts who escaped from certain prisons, only8052were recaptured.

(a) Letprepresent the proportion of all escaped convicts who will eventually be recaptured. Find a point estimate forp. (Round your answer to four decimal places.) (b) Find a 99% confidence interval forp. (Round your answers to three decimal places.)

lower limit
upper limit

Give a brief statement of the meaning of the confidence interval.We are 99% confident that the true proportion of recaptured escaped convicts falls within this interval.We are 99% confident that the true proportion of recaptured escaped convicts falls outside this interval. We are 1% confident that the true proportion of recaptured escaped convicts falls within this interval.We are 1% confident that the true proportion of recaptured escaped convicts falls above this interval.

(c) Is use of the normal approximation to the binomial justified in this problem? Explain.Yes;np< 5 andnq< 5.Yes;np> 5 andnq> 5. No;np< 5 andnq> 5.No;np> 5 andnq< 5.

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For this problem, carry at least four digits after the decimal in your calculations. Answers may vary slightly due to rounding.

A random sample of medical files is used to estimate the proportionpof all people who have blood type B.

(a)

If you have no preliminary estimate forp, how many medical files should you include in a random sample in order to be 85% sure that the point estimatepwill be within a distance of0.05fromp?

Find the critical value for 85% confidence level,

z_0.85,

and find the specified maximal error of estimate,E. (Round your answers to two decimal places.)

z_0.85

=E=

Compute the sample sizenfor estimating the proportionpof all people who have bloodtype Bunder these restrictions. (Round your answer up to the nearest whole number.)

n= medical files

(b)

Answer part (a) if you use the preliminary estimate that about11out of 90 people have blood type B. (Round your proportions to three decimal places and your sample size up to the nearest whole number.)

p=1p=n= medical files

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For this problem, carry at least four digits after the decimal in your calculations. Answers may vary slightly due to rounding.

A random sample of medical files is used to estimate the proportionpof all people who have blood type B.

(a)

If you have no preliminary estimate forp, how many medical files should you include in a random sample in order to be 85% sure that the point estimatepwill be within a distance of0.07fromp? (Round your answer up to the nearest whole number.)

medical files

(b)

Answer part (a) if you use the preliminary estimate that about15out of 90 people have blood type B.

medical files

11.

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For this problem, carry at least four digits after the decimal in your calculations. Answers may vary slightly due to rounding. A random sample of medical files is used to estimate the proportionpof all people who have blood type B.

(a) If you have no preliminary estimate forp, how many medical files should you include in a random sample in order to be 85% sure that the point estimatepwill be within a distance of0.05fromp? (Round your answer up to the nearest whole number.) medical files (b) Answer part (a) if you use the preliminary estimate that about12out of 90 people have blood type B. medical files

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For this problem, carry at least four digits after the decimal in your calculations. Answers may vary slightly due to rounding. How hard is it to reach a businessperson by phone? Letpbe the proportion of calls to businesspeople for which the caller reaches the person being called on thefirsttry.

(a) If you have no preliminary estimate forp, how many business phone calls should you include in a random sample to be 80% sure that the point estimatepwill be within a distance of0.06fromp? (Round your answer up to the nearest whole number.) phone calls (b) A report states that businesspeople can be reached by a single phone call approximately13%of the time. Using this (national) estimate forp, answer part (a). (Round your answer up to the nearest whole number.) phone calls

13.

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For this problem, carry at least four digits after the decimal in your calculations. Answers may vary slightly due to rounding.

What percentage of your campus student body is female? Letpbe the proportion of women students on your campus.

(a)

If no preliminary study is made to estimatep, how large a sample is needed to be 99% sure that a point estimatepwill be within a distance of0.03fromp? (Round your answer up to the nearest whole number.)

students

(b)

A report indicates that approximately58% of college students are females. Answer part (a) using this estimate forp. (Round your answer up to the nearest whole number.)

students