Abottled water distributor wants to estimate the amount of water contained in 1 - gallon bottles purchased from a nationally known water bottling company. the water bottling company's specifications state that the standard deviation of the amount of water is equal to 0.02 gallon. a random sample of 50 bottles is selected, and the sample mean of the water per 1 - gallon bottle is 0.995 gallon.

a. construct a 99% confidence interval estimate for the population mean of water included in a 1 - gallon bottle.

b. on the basis of these results, do you think that the distributor has a right to complain to the water bottling company? why?

c. must you assume that the population amount of water per bottle is normally distributed here? explain.

d. construct a 95% confidence interval estimate. how does this change your answer to (b)?

a)  

The 99% confidence interval is given by (0.988;1.00)  

b) NO, because a 1-gallon of bottle contain exactly 1-gallon of water and this value is inside of the 99% confidence interval.

c) Yes we are assuming this (normality) in order to construct the confidence interval.

d)  

The 95% confidence interval is given by (0.989;1.00)

The confidence interval not changes too much, we got very similar answers.  

Step-by-step explanation:

1) Notation and definitions  

n=50 represent the sample size  

represent the sample mean  

represent the sample standard deviation  

m represent the margin of error  

Confidence =99% or 0.99

A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".  

The margin of error is the range of values below and above the sample statistic in a confidence interval.  

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".  

Part a

In order to find the critical value is important to mention that we know about the population standard deviation, so on this case we need to use the z distribution. Since our interval is at 99% of confidence, our significance level would be given by and .  

We can find the critical values in excel using the following formulas:  

"=NORM.INV(0.005,0,1)" for  

"=NORM.INV(1-0.005,0,1)" for  

The critical value  

The interval for the mean is given by this formula:  

And calculating the limits we got:  

The 99% confidence interval is given by (0.988;1.00)  

Part b

NO, because a 1-gallon of bottle contain exactly 1-gallon of water and this value is inside of the 99% confidence interval.

Part c

Yes we are assuming this (normality) in order to construct the confidence interval.

Part d

In order to find the critical value is important to mention that we know about the population standard deviation, so on this case we need to use the z distribution. Since our interval is at 95% of confidence, our significance level would be given by and .  

We can find the critical values in excel using the following formulas:  

"=NORM.INV(0.025,0,1)" for  

"=NORM.INV(1-0.025,0,1)" for  

The 95% confidence interval is given by (0.989;1.00)

The confidence interval not changes too much, we got very similar answers.  

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