Question

confidence intervals practice
practice problems - do in your notes, answer completely.

1. i randomly select 25 students math sat scores and find \\( \overline{x} = 600 \\). i know that \\( \sigma \\) from this population is 50. find a 95% confidence interval and interpret.
2. i randomly select 25 students math sat scores and find \\( \overline{x} = 600 \\). i know that \\( \sigma \\) from this population is 50. find a 90% confidence interval and interpret.

Explanation:

Step1: Recall the formula for confidence interval

For a population with known standard deviation \(\sigma\), the confidence interval for the population mean \(\mu\) is given by \(\bar{X}\pm z_{\alpha/2}\frac{\sigma}{\sqrt{n}}\)

Step2: Find \(z_{\alpha/2}\) for 95% confidence interval

For 95% confidence interval, \(\alpha = 1 - 0.95=0.05\), \(\alpha/2 = 0.025\). From the standard normal table, \(z_{0.025}=1.96\)

Step3: Calculate the margin of error \(E\)

Given \(\bar{X} = 600\), \(\sigma = 50\), \(n = 25\). The margin of error \(E=z_{\alpha/2}\frac{\sigma}{\sqrt{n}}=1.96\times\frac{50}{\sqrt{25}}=1.96\times10 = 19.6\)

Step4: Construct the confidence interval

The 95% confidence interval is \(\bar{X}-E<\mu<\bar{X} + E\), so \(600 - 19.6<\mu<600+19.6\), which is \(580.4<\mu<619.6\)

Step5: Interpretation

We are 95% confident that the True population mean of Math SAT scores lies between \(580.4\) and \(619.6\)

Step6: Find \(z_{\alpha/2}\) for 90% confidence interval

For 90% confidence interval, \(\alpha=1 - 0.90 = 0.10\), \(\alpha/2=0.05\). From the standard normal table, \(z_{0.05} = 1.645\)

Step7: Calculate the margin of error \(E\)

\(E=z_{\alpha/2}\frac{\sigma}{\sqrt{n}}=1.645\times\frac{50}{\sqrt{25}}=1.645\times10=16.45\)

Step8: Construct the confidence interval

The 90% confidence interval is \(\bar{X}-E<\mu<\bar{X}+E\), so \(600-16.45<\mu<600 + 16.45\), which is \(583.55<\mu<616.45\)

Step9: Interpretation

We are 90% confident that the True population mean of Math SAT scores lies between \(583.55\) and \(616.45\)

Answer:

1. The 95% confidence interval is \(580.4<\mu<619.6\). Interpretation: We are 95% confident that the True population mean of Math SAT scores lies between \(580.4\) and \(619.6\)
2. The 90% confidence interval is \(583.55<\mu<616.45\). Interpretation: We are 90% confident that the True population mean of Math SAT scores lies between \(583.55\) and \(616.45\)