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. A \(90 \%\) confidence interval, \(n=125, \bar{x}=.84\), \(s^{2}=.086\).

Short answer

Answer: The 90% confidence interval for the population mean is approximately \(0.84 \pm 1.656 * \frac{\sqrt{0.086}}{\sqrt{125}}\).

Step-by-step solution

Calculate the Standard Error

To calculate the standard error of the sample mean, we need the sample variance and sample size. The formula for the standard error of the sample mean is: $$SE = \frac{s}{\sqrt{n}}$$ Substituting the given values, we have: $$SE = \frac{\sqrt{0.086}}{\sqrt{125}}$$

Calculate the Degree of Freedom

Before we can find the t-score for the 90% confidence interval, we need to calculate the degree of freedom (DF). The degree of freedom is given by the formula: $$DF = n - 1$$ In this case, $$DF = 125 - 1 = 124$$

Find the 90% Confidence T-Score

Using the degree of freedom, we can find the t-score for the 90% confidence interval in a t-distribution table. Alternatively, you can use a calculator or software to find the value. For a \(90\%\) confidence interval and \(124\) degrees of freedom, the t-score is approximately \(t_{0.05} = 1.656\).

Calculate the Margin of Error

Now that we have the standard error and the t-score, we can calculate the margin of error (ME) for the confidence interval. The formula for the margin of error is: $$ME = t_{0.05} * SE$$ Substitute the earlier calculated values: $$ME = 1.656 * \frac{\sqrt{0.086}}{\sqrt{125}}$$

Calculate the 90% Confidence Interval

Finally, we can calculate the 90% confidence interval for the population mean \(\mu\) using the sample mean \(\bar{x}\) and the margin of error (ME). The confidence interval is given by the formula: $$\bar{x} \pm ME$$ Substitute the given and calculated values: $$0.84 \pm 1.656 * \frac{\sqrt{0.086}}{\sqrt{125}}$$ Compute the lower bound (\(0.84 - ME\)) and upper bound (\(0.84 + ME\)) to get the final 90% confidence interval for the population mean.

Interpret the Confidence Interval

The 90% confidence interval that was computed means that there is a 90% probability that the true population mean \(\mu\) lies within this interval. If we were to sample the population multiple times and compute a \(90\%\) confidence interval each time, we expect that, in the long run, around \(90\%\) of these intervals will contain the true population mean.

Key concepts

Confidence Interval Interpretation

Understanding the interpretation of a confidence interval is a foundational concept in statistics. A confidence interval gives us a range within which we expect the true population parameter, here the mean \(\mu\), to lie with a certain degree of confidence. For instance, when we refer to a \(90\%\) confidence interval, like \(0.84 \pm ME\), we're saying that if we were to take many samples from the population and calculate a \(90\%\) interval for each, we would expect about \(90\%\) of those intervals to contain the true mean \(\mu\).

It's important to highlight that the confidence interval is not a statement about the probability of \(\mu\) being in the interval after it is computed. Instead, it relates to the long-term frequency of how often the true mean would be captured by such intervals across many repetitions of the study. It's a measure of reliability and precision of our estimate rather than a definitive range for \(\mu\).

Standard Error Calculation

Standard error (SE) is a critical statistical concept, serving as an estimate of the variation in the sampling distribution. It informs us how far the sample mean, \(\bar{x}\), is likely to be from the population mean, \(\mu\), should we take another sample. Calculating the standard error is straightforward once you understand it's tied to both the sample's standard deviation (s) and size (n).

The formula is \[ SE = \frac{s}{\sqrt{n}} \. By substituting our sample values, the standard error for our data would be \[ SE = \frac{\sqrt{0.086}}{\sqrt{125}} \]. The smaller the standard error, the more confident we can be in the precision of our sample mean as an estimate of \(\mu\). This calculation becomes even more crucial when paired with the t-score to form confidence intervals.

T-Score for Confidence Interval

A t-score is a standardized score that tells us how many standard errors a point is from the mean of a t-distribution, which becomes important when the sample size is small or the population variance is unknown. The t-score, calculated using degrees of freedom (DF), accounts for variability in the sample.

For our \(90\%\) confidence interval, the t-score - derived from a t-table or statistical software - was found using \(124\) degrees of freedom (based on our sample size of \(125\)) yielding a t-score of approximately \(t_{0.05} = 1.656\). This t-score is then used together with the standard error to calculate the margin of error which bounds the confidence interval:\[ ME = t_{0.05} * SE \] The t-score reflects the confidence level desired (\(90\%\) in our example) and scales the margin of error accordingly. It's a critical factor in accurately determining the range where we are \(90\%\) confident the population mean lies.