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Answer :
To find a 95% confidence interval for the mean amount of caviar per tin, we need to follow a series of steps:
1. Identify the Sample Statistics:
- Sample mean ([tex]\(\bar{x}\)[/tex]) = 99.8 grams
- Sample standard deviation ([tex]\(s\)[/tex]) = 0.9 grams
- Sample size ([tex]\(n\)[/tex]) = 20 tins
2. Determine the Standard Error (SE):
The standard error of the mean is calculated using the formula:
[tex]\[
SE = \frac{s}{\sqrt{n}}
\][/tex]
Substituting the values we have:
[tex]\[
SE = \frac{0.9}{\sqrt{20}} \approx 0.2014
\][/tex]
3. Choose the Appropriate Critical Value:
For a 95% confidence interval with a relatively small sample size (n = 20), we can use the t-distribution critical value. However, the problem provides options that use the z-distribution ([tex]\(1.96\)[/tex]) and an alternative t-distribution value ([tex]\(2.093\)[/tex]). For simplicity and based on the problem choices, we'll proceed with the z-value ([tex]\(1.96\)[/tex]).
4. Calculate the Margin of Error (ME):
Use the critical value and the standard error to find the margin of error:
[tex]\[
ME = \text{critical value} \times SE = 1.96 \times 0.2014 \approx 0.3944
\][/tex]
5. Determine the Confidence Interval:
Now, calculate the confidence interval by adding and subtracting the margin of error from the sample mean:
[tex]\[
\text{Lower limit} = \bar{x} - ME = 99.8 - 0.3944 \approx 99.4056
\][/tex]
[tex]\[
\text{Upper limit} = \bar{x} + ME = 99.8 + 0.3944 \approx 100.1944
\][/tex]
Now, compare the margin of error in each of the given options to the one we calculated.
- Option (A): [tex]\(99.8 \pm 1.96\left(\frac{0.9}{\sqrt{20}}\right)\)[/tex]
The margin of error computed here is [tex]\(0.3944\)[/tex], which matches our calculation.
Therefore, the correct choice is Option (A).
1. Identify the Sample Statistics:
- Sample mean ([tex]\(\bar{x}\)[/tex]) = 99.8 grams
- Sample standard deviation ([tex]\(s\)[/tex]) = 0.9 grams
- Sample size ([tex]\(n\)[/tex]) = 20 tins
2. Determine the Standard Error (SE):
The standard error of the mean is calculated using the formula:
[tex]\[
SE = \frac{s}{\sqrt{n}}
\][/tex]
Substituting the values we have:
[tex]\[
SE = \frac{0.9}{\sqrt{20}} \approx 0.2014
\][/tex]
3. Choose the Appropriate Critical Value:
For a 95% confidence interval with a relatively small sample size (n = 20), we can use the t-distribution critical value. However, the problem provides options that use the z-distribution ([tex]\(1.96\)[/tex]) and an alternative t-distribution value ([tex]\(2.093\)[/tex]). For simplicity and based on the problem choices, we'll proceed with the z-value ([tex]\(1.96\)[/tex]).
4. Calculate the Margin of Error (ME):
Use the critical value and the standard error to find the margin of error:
[tex]\[
ME = \text{critical value} \times SE = 1.96 \times 0.2014 \approx 0.3944
\][/tex]
5. Determine the Confidence Interval:
Now, calculate the confidence interval by adding and subtracting the margin of error from the sample mean:
[tex]\[
\text{Lower limit} = \bar{x} - ME = 99.8 - 0.3944 \approx 99.4056
\][/tex]
[tex]\[
\text{Upper limit} = \bar{x} + ME = 99.8 + 0.3944 \approx 100.1944
\][/tex]
Now, compare the margin of error in each of the given options to the one we calculated.
- Option (A): [tex]\(99.8 \pm 1.96\left(\frac{0.9}{\sqrt{20}}\right)\)[/tex]
The margin of error computed here is [tex]\(0.3944\)[/tex], which matches our calculation.
Therefore, the correct choice is Option (A).
Thank you for reading the article Caviar is an expensive delicacy so companies that package it pay very close attention to the amount of product in their tins Suppose a company. We hope the information provided is useful and helps you understand this topic better. Feel free to explore more helpful content on our website!
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