Question
An exercise equipment company is producing 45 lb. weights. To ensure the quality of their weights, they randomly select 60 weights and find that the mean weight of their sample is 45.07 lbs. with a standard deviation of 0.09 lbs. Create a 95% confidence interval and interpret the answer in context.
(1 point)
Responses
left parenthesis 45.0509 comma 45.0891 right parenthesis; The exercise equipment company can be 95% confident the population mean of their 45 lb. weights lies between 45.0509 and 45.0891 lbs.
Image with alt text: left parenthesis 45.0509 comma 45.0891 right parenthesis ; The exercise equipment company can be 95% confident the population mean of their 45 lb. weights lies between 45.0509 and 45.0891 lbs.
left parenthesis 44.8423 comma 45.2977 right parenthesis; The exercise equipment company can be 95% confident the population mean of their 45 lb. weights lies between 44.8423 and 45.2977 lbs.
Image with alt text: left parenthesis 44.8423 comma 45.2977 right parenthesis ; The exercise equipment company can be 95% confident the population mean of their 45 lb. weights lies between 44.8423 and 45.2977 lbs.
left parenthesis 44.9772 comma 45.0228 right parenthesis; The exercise equipment company can be 95% confident the population mean of their 45 lb. weights lies between 44.9772 and 45.0228 lbs.
Image with alt text: left parenthesis 44.9772 comma 45.0228 right parenthesis ; The exercise equipment company can be 95% confident the population mean of their 45 lb. weights lies between 44.9772 and 45.0228 lbs.
left parenthesis 45.0472 comma 45.0928 right parenthesis; The exercise equipment company can be 95% confident the population mean of their 45 lb. weights lies between 45.0472 and 45.0928 lbs.
(1 point)
Responses
left parenthesis 45.0509 comma 45.0891 right parenthesis; The exercise equipment company can be 95% confident the population mean of their 45 lb. weights lies between 45.0509 and 45.0891 lbs.
Image with alt text: left parenthesis 45.0509 comma 45.0891 right parenthesis ; The exercise equipment company can be 95% confident the population mean of their 45 lb. weights lies between 45.0509 and 45.0891 lbs.
left parenthesis 44.8423 comma 45.2977 right parenthesis; The exercise equipment company can be 95% confident the population mean of their 45 lb. weights lies between 44.8423 and 45.2977 lbs.
Image with alt text: left parenthesis 44.8423 comma 45.2977 right parenthesis ; The exercise equipment company can be 95% confident the population mean of their 45 lb. weights lies between 44.8423 and 45.2977 lbs.
left parenthesis 44.9772 comma 45.0228 right parenthesis; The exercise equipment company can be 95% confident the population mean of their 45 lb. weights lies between 44.9772 and 45.0228 lbs.
Image with alt text: left parenthesis 44.9772 comma 45.0228 right parenthesis ; The exercise equipment company can be 95% confident the population mean of their 45 lb. weights lies between 44.9772 and 45.0228 lbs.
left parenthesis 45.0472 comma 45.0928 right parenthesis; The exercise equipment company can be 95% confident the population mean of their 45 lb. weights lies between 45.0472 and 45.0928 lbs.
Answers
Answered by
GPT-4o mini
To calculate the 95% confidence interval for the mean weight of the weights, we can use the following formula:
\[
\text{Confidence Interval} = \bar{x} \pm z \left( \frac{s}{\sqrt{n}} \right)
\]
where:
- \(\bar{x}\) = sample mean
- \(z\) = z-score corresponding to the desired confidence level (for 95% confidence, \(z \approx 1.96\))
- \(s\) = sample standard deviation
- \(n\) = sample size
Given:
- \(\bar{x} = 45.07\)
- \(s = 0.09\)
- \(n = 60\)
First, we calculate the standard error (SE):
\[
SE = \frac{s}{\sqrt{n}} = \frac{0.09}{\sqrt{60}} \approx \frac{0.09}{7.746} \approx 0.0116
\]
Now, we compute the margin of error (ME):
\[
ME = z \cdot SE = 1.96 \cdot 0.0116 \approx 0.0228
\]
Now we can create the confidence interval:
\[
\text{Confidence Interval} = \bar{x} \pm ME = 45.07 \pm 0.0228
\]
Calculating the lower and upper bounds:
\[
\text{Lower bound} = 45.07 - 0.0228 \approx 45.0472
\]
\[
\text{Upper bound} = 45.07 + 0.0228 \approx 45.0928
\]
Thus, the 95% confidence interval is:
\[
(45.0472, 45.0928)
\]
### Interpretation:
The exercise equipment company can be 95% confident that the population mean of their 45 lb. weights lies between **45.0472 lbs and 45.0928 lbs**.
This means that if they were to take many random samples and compute confidence intervals in the same way, approximately 95% of those intervals would contain the true population mean weight of the weights produced by the company.
\[
\text{Confidence Interval} = \bar{x} \pm z \left( \frac{s}{\sqrt{n}} \right)
\]
where:
- \(\bar{x}\) = sample mean
- \(z\) = z-score corresponding to the desired confidence level (for 95% confidence, \(z \approx 1.96\))
- \(s\) = sample standard deviation
- \(n\) = sample size
Given:
- \(\bar{x} = 45.07\)
- \(s = 0.09\)
- \(n = 60\)
First, we calculate the standard error (SE):
\[
SE = \frac{s}{\sqrt{n}} = \frac{0.09}{\sqrt{60}} \approx \frac{0.09}{7.746} \approx 0.0116
\]
Now, we compute the margin of error (ME):
\[
ME = z \cdot SE = 1.96 \cdot 0.0116 \approx 0.0228
\]
Now we can create the confidence interval:
\[
\text{Confidence Interval} = \bar{x} \pm ME = 45.07 \pm 0.0228
\]
Calculating the lower and upper bounds:
\[
\text{Lower bound} = 45.07 - 0.0228 \approx 45.0472
\]
\[
\text{Upper bound} = 45.07 + 0.0228 \approx 45.0928
\]
Thus, the 95% confidence interval is:
\[
(45.0472, 45.0928)
\]
### Interpretation:
The exercise equipment company can be 95% confident that the population mean of their 45 lb. weights lies between **45.0472 lbs and 45.0928 lbs**.
This means that if they were to take many random samples and compute confidence intervals in the same way, approximately 95% of those intervals would contain the true population mean weight of the weights produced by the company.