Question

A toy manufacturer wants to know how many new toys children buy each year. A sample of 686 children was taken to study their purchasing habits. Construct the 85% confidence interval for the mean number of toys purchase each year if the sample mean was found to be 6.8. Assume that the population standard deviation is 2.1. Round answer to one decimal place

Answer 1

Given that:

- A sample of children taken was:

$n=686$

- The sample mean is:

$\bar{x}=6.8$

- The population standard deviation is:

$\sigma=2.1$

You know that you must construct the 85% confidence interval for the mean number of toys purchased each year.

Then, you need to use the Confidence Interval Formula for the Mean:

$\bar{x}\pm z\cdot\frac{\sigma}{\sqrt{n}}$

Where "z" is the critical value for confidence level, "n" is the sample size, σ is the standard deviation, and this is the sample mean:

$\bar{x}$

By definition the value of "z" for an 85% confidence interval is:

$z=1.44$

Therefore, by substituting values into the formula and evaluating, you get:

$6.8\pm1.44\cdot\frac{2.1}{\sqrt{686}}$$Lower\text{ }endpoint\rightarrow6.8-1.44\cdot\frac{2.1}{\sqrt{686}}\approx6.7$$Upper\text{ }endpoint\rightarrow6.8+1.44\cdot\frac{2.1}{\sqrt{686}}\approx6.9$

Hence, the answer is:

$Lower\text{ }endpoint:6.7$$Upper\text{ }endpoint:6.9$