Ch. 6 - Confidence Intervals

Larson - Elementary Statistics: Picturing the World 8th Edition

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Larson 8th Edition

Ch. 6 - Confidence Intervals

Problem 6.T.3d

Chapter 6, Problem 6.T.3d

The data set represents the scores of 12 randomly selected students on the SAT Physics Subject Test. Assume the population test scores are normally distributed and the population standard deviation is 108. (Adapted from The College Board)

d. Determine the minimum sample size required to be 95% confident that the sample mean test score is within 10 points of the population mean test score.

Verified step by step guidance

Step 1: Identify the formula for minimum sample size calculation. The formula is:

n = (z_{c} σ_{p} / E)^2

, where

z_{c}

is the z-score corresponding to the confidence level,

σ_{p}

is the population standard deviation, and

E

is the margin of error.

Step 2: Determine the z-score for a 95% confidence level. For a 95% confidence level, the z-score is approximately 1.96. This value is derived from standard normal distribution tables.

Step 3: Substitute the given values into the formula. The population standard deviation

σ_{p}

is 108, and the margin of error

E

is 10. Plug these values into the formula:

n = (1.96 108 / 10)^2

Step 4: Simplify the expression inside the parentheses. First, calculate

1.96 108 / 10

. Then square the result to find the value of

n

Step 5: Round up the result to the nearest whole number. Since sample size must be a whole number, always round up to ensure the margin of error is within the specified range.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Sample Size Determination

Sample size determination is the process of calculating the number of observations or replicates needed in a statistical study to ensure that the results are reliable and valid. In this context, it involves using the desired confidence level and margin of error to find the minimum number of students needed to estimate the population mean accurately.

Confidence Interval

A confidence interval is a range of values, derived from a data set, that is likely to contain the true population parameter with a specified level of confidence. For example, a 95% confidence interval means that if we were to take many samples, approximately 95% of those intervals would contain the true population mean, providing a measure of uncertainty around the estimate.

Normal Distribution

Normal distribution is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. In this scenario, the assumption of normality allows the use of specific statistical methods to calculate the sample size and confidence intervals, as many statistical techniques rely on this distribution.

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Related Practice

Textbook Question

Use the standard normal distribution or the t-distribution to construct the indicated confidence interval for the population mean of each data set. Justify your decision. If neither distribution can be used, explain why. Interpret the results.

a. In a random sample of 40 patients, the mean waiting time at a dentist’s office was 20 minutes and the standard deviation was 7.5 minutes. Construct a 95% confidence interval for the population mean.

Textbook Question

The Safe Drinking Water Act, which was passed in 1974, allows the Environmental Protection Agency (EPA) to regulate the levels of contaminants in drinking water. The EPA requires that water utilities give their customers water quality reports annually. These reports include the results of daily water quality monitoring, which is performed to determine whether drinking water is safe for consumption. A water department tests for contaminants at water treatment plants and at customers’ taps. These contaminants include microorganisms, organic chemicals, and inorganic chemicals, such as cyanide. Cyanide’s presence in drinking water is the result of discharges from steel, plastics, and fertilizer factories. For drinking water, the maximum contaminant level of cyanide is 0.2 parts per million. As part of your job for your city’s water department, you are preparing a report that includes an analysis of the results shown in the figure at the right. The figure shows the point estimates for the population mean concentration and the 95% confidence intervals for cyanide over a three-year period. The data are based on random water samples taken by the city’s three water treatment plants.

What can the water department do to decrease the size of the confidence intervals, regardless of the amount of variance in cyanide levels?

Textbook Question

In Exercises 27–30, find the critical values and for the level of confidence c and sample size n.

c = 0.95, n = 13

Textbook Question

b. In a random sample of 15 cereal boxes, the mean weight was 11.89 ounces. Assume the weights of the cereal boxes are normally distributed and the population standard deviation is 0.05 ounce. Construct a 90% confidence interval for the population mean.

Textbook Question

The confidence interval for Year 2 is much larger than that for the other years. What do you think may have caused this larger confidence level?

Textbook Question

In a survey of 2096 U.S. adults, 1740 think football teams of all levels should require players who suffer a head injury to take a set amount of time off from playing to recover. (Adapted from The Harris Poll)

b. Construct a 95% confidence interval for the population proportion. Interpret the results.