Question

Heather runs a successful lawn-mowing business. She would like to estimate the true mean amount of time it takes for her employees to mow a lawn. To do so, she selects a random sample of 30 customers and records the time it takes the employees to mow their lawns. The 90% confidence interval for the true mean time it takes for her employees to mow a lawn is 40–55 minutes. Which of the following would decrease the width of the interval?

Answer 1

The width of the confidence interval can be decreased by either increasing the sample size or reducing the level of confidence.

Step-by-step explanation:

In statistical terms, the width of a confidence interval is influenced by two main factors: the standard error and the critical value. The standard error is inversely proportional to the square root of the sample size (n), while the critical value is determined by the level of confidence. Mathematically, the formula for the confidence interval is given by:

`\[ \text{Confidence Interval} = \text{Sample Mean} \pm (\text{Critical Value} * \text{Standard Error}) \]`

For a given sample size, increasing the critical value (associated with a higher level of confidence) will widen the interval. Similarly, increasing the sample size will decrease the standard error and, consequently, narrow the interval.

In Heather's case, if she wants to decrease the width of the confidence interval (40–55 minutes), she can consider either reducing the level of confidence (which corresponds to a lower critical value) or increasing the sample size.

For example, switching from a 90% confidence interval to an 80% confidence interval will result in a narrower range, but it comes at the cost of being less certain about the estimate. Alternatively, if feasible, Heather could collect data from a larger number of customers to achieve a more precise estimate of the mean mowing time.

Answer 2

The true mean time it takes for her employees to mow a lawn would be captured by about 90% of the constructed intervals if numerous random samples of size 30 are chosen from the population of all customers.

Thanks.