Answer:
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.