Questions: Use the t-distribution to find a confidence interval for a mean μ given the relevant sample results. Give the best point estimate for μ, the margin of error; and the confidence interval. Assume the results come from a random sample from a population that is approximately normally distributed. A 95% confidence interval for μ using the sample results x̄=87.0, s=7.2, and n=42 Round your answer for the point estimate to one decimal place, and your answers for the margin of error and the confidence interval to two decimal places. point estimate = margin of error = The 95% confidence interval is to .

Use the t-distribution to find a confidence interval for a mean μ given the relevant sample results. Give the best point estimate for μ, the margin of error; and the confidence interval. Assume the results come from a random sample from a population that is approximately normally distributed.

A 95% confidence interval for μ using the sample results x̄=87.0, s=7.2, and n=42
Round your answer for the point estimate to one decimal place, and your answers for the margin of error and the confidence interval to two decimal places.
point estimate = 
margin of error = 

The 95% confidence interval is  to  .
Transcript text: Current Attempt in Progress Use the $t$-distribution to find a confidence interval for a mean $\mu$ given the relevant sample results. Give the best point estimate for $\mu$, the margin of error; and the confidence interval. Assume the results come from a random sample from a population that is approximately normally distributed. A 95\% confidence interval for $\mu$ using the sample results $\bar{x}=87.0, s=7.2$, and $n=42$ Round your answer for the point estimate to one decimal place, and your answers for the margin of error and the confidence interval to two decimal places. point estimate $=$ $\square$ margin of error = $\square$ The 95\% confidence interval is $\square$ to $\square$ . eTextbook and Media
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Solution

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Solution Steps

Step 1: Point Estimate

The best point estimate for the population mean μ \mu is given by the sample mean xˉ \bar{x} : Point estimate=xˉ=87.0 \text{Point estimate} = \bar{x} = 87.0

Step 2: Margin of Error

To calculate the margin of error, we use the formula: Margin of Error=Zsn \text{Margin of Error} = Z \cdot \frac{s}{\sqrt{n}} where Z Z is the Z-score corresponding to the 95% confidence level, s s is the sample standard deviation, and n n is the sample size. For a 95% confidence level, Z=1.96 Z = 1.96 .

Substituting the values: Margin of Error=1.967.2422.18 \text{Margin of Error} = 1.96 \cdot \frac{7.2}{\sqrt{42}} \approx 2.18

Step 3: Confidence Interval

The confidence interval for the mean is calculated as: xˉ±Margin of Error \bar{x} \pm \text{Margin of Error} Thus, we have: 87.0±2.18 87.0 \pm 2.18 Calculating the lower and upper bounds: Lower Bound=87.02.18=84.82 \text{Lower Bound} = 87.0 - 2.18 = 84.82 Upper Bound=87.0+2.18=89.18 \text{Upper Bound} = 87.0 + 2.18 = 89.18 Therefore, the 95% confidence interval is: (84.82,89.18) (84.82, 89.18)

Final Answer

  • Point estimate =87.0 = 87.0
  • Margin of error =2.18 = 2.18
  • The 95% confidence interval is from 84.82 84.82 to 89.18 89.18 .

Point estimate=87.0, Margin of error=2.18, Confidence interval=(84.82,89.18) \boxed{\text{Point estimate} = 87.0, \text{ Margin of error} = 2.18, \text{ Confidence interval} = (84.82, 89.18)}

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