Question

In a survey of 35 adult Americans, it was found that the mean age (in years) that people would like to live to is 87.9 with a standard deviation of 15.5.

a) Explain how a large sample size is useful in constructing a confidence interval for the mean age people would like to live.
b) Construct and interpret a 95% confidence interval for the mean age that people would like to live.

Answer 1

a) A larger sample size in a survey provides a more accurate estimate of the population mean age people would like to live to. It reduces sampling variability and increases the precision of the confidence interval constructed for the mean age.

b) The 95% confidence interval for the mean age that people would like to live is
`\(\bar{x} \pm Z * (s)/(√(n)) = 87.9 \pm 1.96 * (15.5)/(√(35))\),` which calculates to approximately 82.9 to 92.9 years. This means we are 95% confident that the true mean age people would like to live falls between these values.

Step-by-step explanation:

a) A larger sample size is beneficial in constructing a confidence interval as it reduces the standard error of the mean. A smaller standard error results in a narrower confidence interval, providing a more precise estimate of the population parameter. With a larger sample size, the variability within the sample better represents the variability of the entire population, reducing the chances of a sampling error and increasing the reliability of the estimate.

b) To construct a 95% confidence interval for the mean age people would like to live, we use the formula
`\(\bar{x} \pm Z * (s)/(√(n))\).`Given a mean age of 87.9 years
`(\(\bar{x}\))`, a standard deviation of 15.5 years (s), and a sample size of 35 (n), the critical value at a 95% confidence level (Z) is 1.96 (obtained from the standard normal distribution). Calculating the interval yields 87.9 ± 1.96 × (15.5 / √35), resulting in a range of approximately 82.9 to 92.9 years. This means that if we were to take multiple samples and construct confidence intervals in this manner, we would expect 95% of those intervals to contain the true population mean age people would like to live.