Long Life? 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. An analysis of the raw data indicates the distribution is skewed left.


b. Construct and interpret a 95% confidence interval for the mean.

Verified step by step guidance

1. Identify the sample size \(n = 35\), the sample mean \(\bar{x} = 87.9\), and the sample standard deviation \(s = 15.5\). Since the population standard deviation is unknown and the sample size is less than 30, we will use the t-distribution to construct the confidence interval.
2. Determine the degrees of freedom, which is \(df = n - 1 = 35 - 1 = 34\).
3. Find the critical t-value \(t^*\) for a 95% confidence level and 34 degrees of freedom. This value can be found using a t-distribution table or statistical software by looking up the two-tailed value corresponding to \(\alpha = 0.05\).
4. Calculate the standard error of the mean (SEM) using the formula: \(\text{SEM} = \frac{s}{\sqrt{n}} = \frac{15.5}{\sqrt{35}}\)
5. Construct the 95% confidence interval using the formula: \(\bar{x} \pm t^* \times \text{SEM}\) This interval estimates the range in which the true population mean age that people would like to live to lies with 95% confidence. Interpret the interval by explaining that we are 95% confident the true mean falls within this range.

Key Concepts

Confidence Interval for the Mean

A confidence interval estimates the range within which the true population mean is likely to fall, based on sample data. For a 95% confidence level, we expect that 95% of such intervals calculated from repeated samples will contain the true mean. It combines the sample mean, variability, and sample size to quantify uncertainty.

t-Distribution

When the population standard deviation is unknown and the sample size is small (typically n < 30 or 40), the t-distribution is used instead of the normal distribution. It accounts for extra uncertainty by having heavier tails, and its shape depends on degrees of freedom (sample size minus one).

Effect of Skewness on Confidence Intervals

Skewness indicates asymmetry in data distribution. A left-skewed distribution means data has a longer tail on the left. While the t-interval assumes approximate normality, moderate skewness can affect accuracy, especially with small samples, so interpretation should consider this limitation.