Statistics: Confidence Intervals and Margin of Error Worksheet

Question 1

A survey of 200 students found that 124 prefer online homework. Find the sample proportion of students who prefer online homework.

Accepted Answer

The sample proportion is 124 divided by 200, which equals 0.62. This means 62% of the students in the sample prefer online homework.

Question 2

A poll reports that 48% of voters support a proposal, with a margin of error of 3 percentage points. Write the confidence interval for the true percent of voters who support the proposal.

Accepted Answer

The confidence interval is 45% to 51%. This comes from subtracting and adding 3 percentage points to 48%.

Question 3

A confidence interval for the average number of minutes students spend reading each night is 22 minutes to 34 minutes. Find the margin of error.

Accepted Answer

The margin of error is 6 minutes. The midpoint is 28 minutes, and each endpoint is 6 minutes away from 28.

Question 4

A 95% confidence interval for the mean height of a certain plant species is 18.4 cm to 21.6 cm. Write a correct interpretation of this interval.

Accepted Answer

We are 95% confident that the true mean height of this plant species is between 18.4 cm and 21.6 cm.

Question 5

A sample mean is 72 and the margin of error is 5. Write the confidence interval.

Accepted Answer

The confidence interval is 67 to 77. This is found by subtracting 5 from 72 and adding 5 to 72.

Question 6

A news report says, "The survey has a margin of error of plus or minus 4%." Explain what this means in plain language.

Accepted Answer

It means the reported sample percent may be about 4 percentage points above or below the true population percent because of sampling variability.

Question 7

A researcher wants a smaller margin of error when estimating a population proportion. Should the researcher use a larger sample size or a smaller sample size? Explain.

Accepted Answer

The researcher should use a larger sample size. Larger samples usually reduce sampling variability, which makes the margin of error smaller.

Question 8

A 90% confidence interval for the proportion of teens who have a part-time job is 0.28 to 0.36. Find the point estimate.

Accepted Answer

The point estimate is 0.32. It is the midpoint of the interval because 0.28 and 0.36 are each 0.04 away from 0.32.

Question 9

A sample of 64 batteries has a mean life of 40 hours. The standard deviation is known to be 8 hours. Using a 95% confidence level with z = 1.96, find the margin of error. Use the formula margin of error = z times standard deviation divided by square root of n.

Accepted Answer

The margin of error is 1.96 times 8 divided by the square root of 64. Since the square root of 64 is 8, the margin of error is 1.96 hours.

Question 10

Using the information from the battery problem, write the 95% confidence interval for the true mean battery life.

Accepted Answer

The confidence interval is 38.04 hours to 41.96 hours. This is found by subtracting and adding the margin of error, 1.96 hours, to the sample mean of 40 hours.

Question 11

Two surveys estimate the same population proportion. Survey A has a confidence interval of 52% to 58%. Survey B has a confidence interval of 49% to 61%. Which survey has the smaller margin of error? Explain.

Accepted Answer

Survey A has the smaller margin of error. Survey A has a margin of error of 3 percentage points, while Survey B has a margin of error of 6 percentage points.

Question 12

A student says, "A 95% confidence interval means there is a 95% chance that the sample mean is in the interval." Explain why this statement is not correct.

Accepted Answer

The statement is not correct because the sample mean is the center of the interval, so it is already in the interval. A better interpretation is that we are 95% confident the interval captures the true population mean.

Statistics: Confidence Intervals and Margin of Error

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