# Statistics Help - Confidence Interval

• Absolut
In summary, Builder A took a sample of size n = 4000, and Builder B took a sample of size n = 40. The variances of the samples are equal, and using the formula s/sqrt(n), the sizes of the samples can be determined to be n = 4000 for Builder A and n = 40 for Builder B. This is based on the given confidence intervals and assuming no errors in calculations were made.

#### Absolut

I*m looking for some help with a statistics question:
Builder A and Builder B both take random samples of nails, and the width of the nails are measured in centimetres - Builder A estimates that a 95% confidence interval for the width of the nails is 1.2 +/- 1.96 * 0.05. Builder B estimates that a 95% confidence interval for the width of the nails is 1.2 +/- 1.96 * 0.005. Neither builder had knowledge of the population variance of the widths of the nauks, but the variance of the samples are equal. Assuming that no errors in calculations are made, find the size of Builder As sample and Builder Bs sample.

It's a MCQ - with options as follows:
A) Builder A took a sample of size n = 40, and Builder B took a sample of size n = 4000
B) Builder A took a sample of size n = 4000, and Builder B took a sample of size n = 40
C) Builder A took a sample of size n = 10, and Builder B took a sample of size n = 1000
D) Builder A took a sample of size n = 1000, and Builder B took a sample of size n = 10
E) Builder A took a sample of size n = 100, and Builder B took a sample of size n = 100.

Since the variances are equal, I thought that maybe s/rootx would be equal to s/rooty, for x equal to the size of Builder As sample and y equal to the size of Builder Bs sample, but this turned out to be wrong - since this method yields two possible answers.

Any help is appreciated.

You are close. The s/sqrt(n_x) and s/sqrt(n_y) are not equal, but they are the numbers that multiply the 1.96 factor in each case

The correct answer is D) Builder A took a sample of size n = 1000, and Builder B took a sample of size n = 10.

To understand why this is the correct answer, let's first break down the information given in the question. Both Builder A and Builder B have taken random samples of nails and measured their width in centimeters. They both estimate a 95% confidence interval for the width of the nails, with Builder A's interval being 1.2 +/- 1.96 * 0.05 and Builder B's interval being 1.2 +/- 1.96 * 0.005. This means that Builder A's estimate has a margin of error of 0.05 cm and Builder B's estimate has a margin of error of 0.005 cm.

Now, let's look at the options given. A) Builder A took a sample of size n = 40, and Builder B took a sample of size n = 4000. This cannot be the correct answer because the margin of error for Builder A's estimate is much larger than the margin of error for Builder B's estimate. This would mean that Builder A's sample size should be smaller, not larger, than Builder B's sample size.

B) Builder A took a sample of size n = 4000, and Builder B took a sample of size n = 40. This option also cannot be correct for the same reason as option A.

C) Builder A took a sample of size n = 10, and Builder B took a sample of size n = 1000. This option also cannot be correct because the margin of error for Builder A's estimate is much larger than the margin of error for Builder B's estimate. This would mean that Builder A's sample size should be smaller, not larger, than Builder B's sample size.

E) Builder A took a sample of size n = 100, and Builder B took a sample of size n = 100. This option also cannot be correct because the margin of error for Builder A's estimate is the same as the margin of error for Builder B's estimate. This would mean that both samples should have the same size, which is not the case according to the given information.

This leaves us with option D) Builder A took a sample of size n = 1000, and Builder B took a sample of size n = 10. This is the correct answer because the

## 1. What is a confidence interval?

A confidence interval is a range of values that is likely to include the true value of a population parameter with a certain level of confidence. It is based on a sample of data and is used to estimate the true value of a population parameter.

## 2. How is a confidence interval calculated?

A confidence interval is calculated by taking the sample mean and adding and subtracting the margin of error to it. The margin of error is determined by the level of confidence and the standard deviation of the sample.

## 3. What is the significance of the level of confidence in a confidence interval?

The level of confidence represents the probability that the true population parameter falls within the calculated confidence interval. For example, a 95% confidence interval means that there is a 95% chance that the true population parameter falls within the interval.

## 4. What factors can affect the width of a confidence interval?

The width of a confidence interval is affected by the sample size, the standard deviation of the sample, and the level of confidence. A larger sample size and a smaller standard deviation will result in a narrower confidence interval.

## 5. How can confidence intervals be used in statistical analysis?

Confidence intervals are used to estimate the true value of a population parameter and to determine the precision of the estimate. They can also be used to compare the means of two populations and to test hypotheses about the true value of a population parameter.