The data set represents the weights (in pounds) of 10 randomly selected black bears from northeast Pennsylvania. Assume the weights are normally distributed. (Source: Pennsylvania Game Commission)

b. Construct a 95% confidence interval for the population mean. Interpret the results.

Verified step by step guidance

1. Step 1: Calculate the sample mean (x̄) by summing all the weights in the data set and dividing by the number of observations (n = 10). Use the formula: $x¯=\frac{\sum x}{n}$.
2. Step 2: Calculate the sample standard deviation (s) using the formula: $s=\sqrt{\frac{\sum (x-x¯{)}^2}{n-1}}$. This involves finding the squared differences between each weight and the sample mean, summing them, dividing by (n-1), and taking the square root.
3. Step 3: Determine the critical value (t*) for a 95% confidence interval using a t-distribution table. Since the sample size is 10, the degrees of freedom (df) will be n-1 = 9. Look up the t* value corresponding to df = 9 and a 95% confidence level.
4. Step 4: Calculate the margin of error (E) using the formula: $E=t*\frac{s}{\sqrt{n}}$. Plug in the values for t*, s, and n.
5. Step 5: Construct the confidence interval for the population mean using the formula: $[x¯-E,x¯+E]$. Interpret the results by explaining that the interval provides a range of plausible values for the population mean weight of black bears with 95% confidence.

Key Concepts

Normal Distribution

Normal distribution is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. It is characterized by its bell-shaped curve, defined by its mean and standard deviation. In this context, assuming the weights of the bears are normally distributed allows for the application of statistical methods, such as confidence intervals.

Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the value of an unknown population parameter. For a 95% confidence interval, we can say that if we were to take many samples and build intervals, approximately 95% of those intervals would contain the true population mean. This concept is crucial for estimating the mean weight of the bear population based on the sample data.

Sample Mean and Standard Deviation

The sample mean is the average of a set of values, calculated by summing all the observations and dividing by the number of observations. The standard deviation measures the amount of variation or dispersion in a set of values. In constructing a confidence interval, both the sample mean and standard deviation are used to determine the range within which the population mean is likely to fall.