7.4 Confidence Intervals
Chapter 7
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Solutions 17 Videos

Common confidence levels are 90%, 95%, and 99%. A sample is taken from a population with a given mean and standard deviation. Within approximately how many standard deviations of the mean will the values in the confidence interval lie for a

a) 90% confidence level?

b) 95% confidence level?

c) 99% confidence level?

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A poll conducted by the student newspaper found that 78% of the students who ate lunch at the school cafeteria ordered a salad at least twice per week. The poll is considered accurate within _+5%, 17 times out of 20. What is the confidence level for the poll?

A 75%

B 85%

C 95%

D 98%

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During a municipal election the local newspaper polled 251 people. The paper reported that 57% said they were in favour of candidate A for mayor. The result was considered accurate within 6.1%, 19 times out of 20. Which of the following statements is false?

A The margin of error is 6.1%.

B In a similar poll, 95% of the time between 50.9% and 63.1% of the people would be found in favour.

C The confidence interval is 57% \pm 6.1%.

D In a similar poll, 95% of the time 57% or more of the people would be found in favour.

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An automobile dealer offers a new line of tires. The tires last a mean of 75 000 km with a standard deviation of 5000 km, following a normal distribution. The tire life experienced by 100 customers is recorded. What is the expected standard deviation of the sample means?

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The Canadian commercial pilot written exam consists of 100 multiple-choice questions. Last year, the students enrolled in a community college aviation course recorded a mean mark of 82% among 25 candidates.

a) Determine the margin of error at a 99% confidence level.

b) Determine the confidence interval for the exam marks.

c) State the results in the usual format for the course newsletter.

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A political party received an average of 34% support in recent polls plus or minus 3.4%”. 19 times out of 20. Two subsequent polls showed 38% support and 27% support. How would you report on the meaning of these polls to the party membership?

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A Single Cre‘me cookie is made using a cream filling between two wafters. The amount of cream follows a normal distribution with a mean of 25 g and a standard deflation of 2.0 g. The company claims its new Double Cre‘me line contains twice the amount of filling. A random sample of 20 such cookies were found to contain cream content as shown.

a) Calculate the mean of the sample and the standard deviation for the sample means. What assumption must you make?

b) Determine the 95% confidence interval for the sample mean.

c) Is the company justified in claiming that the Double Creme line contains twice the filling of the Single Creme line? Give reasons for your answer.

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Aconcretemanufacturerkl'lO\VSfrom experience that setting times for concrete follow a normal distribution with a standard deviation of a = 8.5 min. The manufacturer wants to use the slogan “Our concrete quick~sets in t minutes” in its advertising campaign. A technician pours 25 test squares of equal size and finds the mean setting time to be 72.2 min.

a) Determine a 95% confidence interval for the actual mean setting time of the concrete.

b) Advise the manufacturer on a reasonable value for t. Give a reason for your answer.

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A honey farm rates its honey on a colour scale from 1 to 20, ranging from very light orange to deep orange. The colour of honey follows a normal distribution with a standard deviation of 2.5. A technician tests a sample of 50 jars of honey, resulting in a sample mean of 12. Determine a 95% confidence interval for the colour of the honey.

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A survey of businesses showed that a mean of 17% of gross income was spent on office overhead. with a standard deviation of 5%, following a normal distribution. At a 99% confidence level, the margin of error was 10.5%. How many businesses were surveyed?

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As part of Earth Day celebrations, an environmental scientist participated in' a program to measure water clarity m' 70 locations in Lake Ontario using a clarity measuring disk. The scientist reported that the lake water was clear to a mean depth of 5 m with a standard deviation of 1 m. The margin of error was 0.20 m. What confidence level was used?

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A study of patients with low-back pain reported that the sample mean duration of the pain was 18.3 months. The duration follows a normal distribution in the population with a standard deviation of 5.9 months. The margin of error for the population mean was 1.5 months at a 95% confidence level. How many patients were in the study?

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A manufacturer of computer hardware knows that the life of its hard drives follows a normal distribution with a standard deviation of 2400 h. Over the past three years, the mean life, based on a sample of 900 hard drives, was 16 000 h of use. What is the 95% confidence interval for the mean life of a hard drive?

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When tomato farmers harvest their crop, they use an automated tomato picker that separates the tomato from the vine and eliminates any non—ripe tomatoes. The process of eliminating the non-ripe tomatoes is not perfect due to the speed with which the tomatoes are actually picked. Some green tomatoes get into the load. The buyer takes a single scoop of tomatoes from a random spot in the load. A sample of 300 tomatoes contains 92% of an acceptable standard.

a) What is the confidence interval of the load if the confidence level is 95%?

b) The current load measures 41.992 tonnes. What is the interval for the mass of acceptable tomatoes?

c) Tomatoes sell for $94.40 per tonne. What is the range of values that this load is worth?

d) A load is accepted as long as the sample contains at least 67% acceptable tomatoes. How does the range of values change for a load at this level?

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A friend shows you a caption from the school newspaper: “Two recent polls show that 57% of students would vote for Adam and 51% would vote for Meghan in the next student council election. These results are accurate to within \pm 3%, 19 times in 20."

Explain what the caption means.

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Suppose you select 30 men from the entire population of 20-year-old men, and measure their weights.

a) Are you more likely to end up with men close to the population mean or far away from the population mean? Give a reason for your answer.

b) What effect does this have on the standard deviation of the sample mean compared to the population mean?

c) How does the formula for the standard deviation of the sample means, \sigma_x = \frac{\sigma}{\sqrt{n}}, fit with the effect in part b)?

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Consider the formula for the margin of error:

E = z \sqrt{\frac{p(1-p)}{n}}, A poll is being confucted

to determine the percent of voters who w111' vote for candidate Alpha with a confidence level of 95%. What is the minimum number of voters who must be interviewed to guarantee a margin of error of no more than 2%, regardless of the value of p? Explain your reasoning.

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