Question

Use the information given in Exercises 9-15 to find the necessary confidence interval for the population mean \(\mu .\) Interpret the interval that you have constructed. \(\alpha=.05, n=89, \bar{x}=66.3, s^{2}=2.48\)

Short answer

Based on the given data and calculations, the 95% confidence interval for the population mean is (65.968, 66.632). This indicates that we are 95% confident that the true population mean lies within this range.

Step-by-step solution

Identify given values and the formula for a confidence interval

We are given the following data: - Level of significance (\(\alpha\)): 0.05 - Sample size (n): 89 - Sample mean (\(\bar{x}\)): 66.3 - Sample variance (\(s^2\)): 2.48 The formula for a confidence interval for the population mean, when the population standard deviation is unknown, is given by: \(\bar{x} \pm t \dfrac{s}{\sqrt{n}}\)

Calculate the degrees of freedom and the t-score

First, we need to calculate the degrees of freedom (df) for the t-distribution, which is given by: df = n - 1 = 89 - 1 = 88 Now, using a t-table or an online calculator, we find the t-score for the 95% confidence level and 88 degrees of freedom: \(t_{\alpha/2, 88} = 1.989\)

Calculate the standard error

Next, we need to find the standard error of the mean, which is given by: \(SE = \dfrac{s}{\sqrt{n}}\) Given \(s^2 = 2.48\), the sample standard deviation is: \(s = \sqrt{2.48} = 1.5748\) Then, the standard error is: \(SE = \dfrac{1.5748}{\sqrt{89}} \approx 0.167\)

Calculate the confidence interval

Now, we can find the confidence interval using the formula we identified earlier: \(\bar{x} \pm t_{\alpha/2, df} \cdot SE\) \(66.3 \pm 1.989 \cdot 0.167\) \(66.3 \pm 0.332\) Theconfidence interval is \((65.968, 66.632)\).

Interpret the confidence interval

The constructed 95% confidence interval for the population mean \(\mu\) is \((65.968, 66.632)\). This means that we can be 95% confident that the true population mean lies within this interval.

Key concepts

Population Mean Estimation

Estimating the population mean is a fundamental aspect of statistics, particularly when we strive to infer the average of a whole population based on a sample. In the absence of full population data, we take a representative sample and calculate the sample mean \( \bar{x} \). From here, we utilize the sample mean to estimate the unknown population mean \( \mu \).

The challenge is that our sample could be one of many possible samples we might have drawn from the population, and each sample could have a different sample mean. Thus, we calculate a confidence interval to express the range within which the true population mean likely resides, considering a certain level of confidence (often 95% or 99%). The confidence interval uses the sample mean as the center point and adds and subtracts a margin of error to create the range.

This margin of error is determined by the standard error of the mean and a multiplier sourced from a distribution that reflects the uncertainty in our estimate – for smaller sample sizes or unknown population standard deviation, this multiplier comes from the t-distribution.

t-Distribution

The t-distribution, or Student's t-distribution, is a probability distribution that arises when estimating the mean of a normally distributed population in situations where the sample size is small and the population standard deviation is unknown.

It is similar to the standard normal distribution, but with thicker tails, which means it predicts a greater likelihood of values further from the mean. The shape of the t-distribution changes with the sample size, becoming more similar to the normal distribution as the sample size increases.

The 't-score' is a scaled estimate of the difference between the sample mean and the population mean relative to the standard error. To use the t-distribution for calculating the confidence interval, we must find the t-score that corresponds to our desired confidence level. For instance, a 95% confidence level means we want the middle 95% of the t-distribution; the t-score is the value such that 2.5% of the distribution lies to its right and left respectively.

Degrees of Freedom

Degrees of freedom (df) are a crucial concept when using the t-distribution to estimate a population parameter like the mean. The term 'degrees of freedom' refers to the number of independent values in a sample that can vary when estimating a statistical parameter.

When calculating a confidence interval for the population mean using a t-distribution, we determine the degrees of freedom as the sample size minus one \( n-1 \). This subtraction accounts for the fact that we are using the sample to estimate the population standard deviation. The larger the degrees of freedom, the closer our t-distribution will resemble the standard normal distribution.

In our exercise, with a sample size of 89, we have 88 degrees of freedom. With these degrees of freedom, we can identify the correct t-score from the t-distribution table or an online calculator, which is then used, along with the standard error, to calculate the range of the confidence interval.