A paint manufacturer uses a machine to fill gallon cans with paint (1 gal = 128 ounces). The manufacturer wants to estimate the mean volume of paint the machine is putting in the cans within 0.6 ounce. Assume the population of volumes is normally distributed. (a) Determine the minimum sample size required to construct a 90% confidence interval for the population mean. Assume the population standard deviation is 0.70 ounce. (b) The sample mean is 126.25 ounces. With a sample size`of 6, a 90% level of confidence, and a population standard deviation of 0.70 ounce, does it seem possible that the population mean could be exactly 128 ounces? Explain. Click here to view page 1 of the Standard Normal Table. Click here to view page 2 of the Standard Normal Table, (a) The minimum sample size required to construct a 90% confidence interval is cans. (Round up to the nearest whole number.) (b) The 90% confidence interval for a sample size of 6 is ). It does seem possible that the population mean could be exactly 128 ounces because 128 ounces falls inside the confidence interval. (Round to two decimal places as needed.)

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### Estimating the Minimum Sample Size and Constructing Confidence Intervals

A paint manufacturer utilizes a machine to fill gallon cans with paint (1 gal = 128 ounces). The manufacturer wants to estimate the mean volume of paint the machine is putting in the cans within 0.6 ounces. Assume the population of volumes is normally distributed. 

#### Tasks:

**(a)** Determine the minimum sample size required to construct a 90% confidence interval for the population mean. Assume the population standard deviation is 0.70 ounce.

**(b)** The sample mean is 126.25 ounces. With a sample size of 6, a 90% level of confidence, and a population standard deviation of 0.70 ounce, does it seem possible that the population mean could be exactly 128 ounces? Explain.

**References:**
<a href="https://www.sjsu.edu/faculty/gerstman/StatPrimer/z-table.pdf" target="_blank">Click here to view page 1 of the Standard Normal Table.</a>
<a href="https://www.sjsu.edu/faculty/gerstman/StatPrimer/z-table.pdf" target="_blank">Click here to view page 2 of the Standard Normal Table.</a>

---

#### Solutions:

**(a)** The minimum sample size required to construct a 90% confidence interval is __[number]__ cans.

_(Round up to the nearest whole number)_

**(b)** The 90% confidence interval for a sample size of 6 is (__[lower bound]__, __[upper bound]__). It __[does/doesn't]__ seem possible that the population mean could be exactly 128 ounces because 128 ounces falls __[inside/outside]__ the confidence interval.

_(Round to two decimal places as needed.)_

---

Please use the [Standard Normal Table](https://www.sjsu.edu/faculty/gerstman/StatPrimer/z-table.pdf) to find the z-values needed for these calculations. Familiarity with using the Standard Normal Table, understanding of confidence intervals, and sample size determination are essential for solving these problems effectively.
Transcribed Image Text:### Estimating the Minimum Sample Size and Constructing Confidence Intervals A paint manufacturer utilizes a machine to fill gallon cans with paint (1 gal = 128 ounces). The manufacturer wants to estimate the mean volume of paint the machine is putting in the cans within 0.6 ounces. Assume the population of volumes is normally distributed. #### Tasks: **(a)** Determine the minimum sample size required to construct a 90% confidence interval for the population mean. Assume the population standard deviation is 0.70 ounce. **(b)** The sample mean is 126.25 ounces. With a sample size of 6, a 90% level of confidence, and a population standard deviation of 0.70 ounce, does it seem possible that the population mean could be exactly 128 ounces? Explain. **References:** <a href="https://www.sjsu.edu/faculty/gerstman/StatPrimer/z-table.pdf" target="_blank">Click here to view page 1 of the Standard Normal Table.</a> <a href="https://www.sjsu.edu/faculty/gerstman/StatPrimer/z-table.pdf" target="_blank">Click here to view page 2 of the Standard Normal Table.</a> --- #### Solutions: **(a)** The minimum sample size required to construct a 90% confidence interval is __[number]__ cans. _(Round up to the nearest whole number)_ **(b)** The 90% confidence interval for a sample size of 6 is (__[lower bound]__, __[upper bound]__). It __[does/doesn't]__ seem possible that the population mean could be exactly 128 ounces because 128 ounces falls __[inside/outside]__ the confidence interval. _(Round to two decimal places as needed.)_ --- Please use the [Standard Normal Table](https://www.sjsu.edu/faculty/gerstman/StatPrimer/z-table.pdf) to find the z-values needed for these calculations. Familiarity with using the Standard Normal Table, understanding of confidence intervals, and sample size determination are essential for solving these problems effectively.
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