Question
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a. Identify the critical value \( t_{\alpha / 2} \) used for finding the margin of error. \( t_{\alpha / 2}=1.69 \) (Round to two decimal places as needed.) b. Find the margin of error. \( \mathrm{E}=195.9 \mathrm{~g} \) (Round to one decimal place as needed.) c. Find the confidence interval estimate of \( \mu \). \( \square \mathrm{g}<\mu<\square \mathrm{g} \) (Round to one decimal place as needed.)

Ask by Kelly Warner. in the United States
Nov 06,2024

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Tutor-Verified Answer

Answer

The confidence interval estimate of \( \mu \) is \( \bar{x} - 195.9 < \mu < \bar{x} + 195.9 \).

Solution

To find the confidence interval estimate of \( \mu \), we can use the following steps: ### a. Critical Value You have already identified the critical value: \[ t_{\alpha / 2} = 1.69 \] ### b. Margin of Error You have also provided the margin of error: \[ E = 195.9 \, \text{g} \] ### c. Confidence Interval Estimate of \( \mu \) To find the confidence interval estimate of \( \mu \), we need the sample mean (\( \bar{x} \)) and the margin of error (\( E \)). The confidence interval is calculated as follows: \[ \text{Lower limit} = \bar{x} - E \] \[ \text{Upper limit} = \bar{x} + E \] However, since you haven't provided the sample mean (\( \bar{x} \)), I will denote it as \( \bar{x} \) for now. Thus, the confidence interval can be expressed as: \[ \bar{x} - 195.9 < \mu < \bar{x} + 195.9 \] If you provide the sample mean, I can calculate the specific values for the confidence interval. Otherwise, the confidence interval in terms of \( \bar{x} \) is: \[ \bar{x} - 195.9 < \mu < \bar{x} + 195.9 \] Please provide the sample mean to complete the calculation.

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The Deep Dive

To find the confidence interval estimate of \( \mu \), you can follow this fun recipe: Start with the sample mean (which we'll assume you know), and then simply subtract and add the margin of error \( E \) (195.9 g) to this value. Let's say your sample mean is 500 g. The confidence interval would be 500 g - 195.9 g < \( \mu \) < 500 g + 195.9 g, giving us 304.1 g < \( \mu \) < 695.9 g. Now, remember, confidence intervals are wonderful! They provide a range of values that we believe contains the true population mean \( \mu \), with a certain level of confidence. This means that if you were to take many samples and create intervals, a significant number of them would capture the true mean. Keep this in mind when interpreting your results—it's all about giving you a better understanding of your data!

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