a) There is no significant evidence to conclude that there there is significant difference in the mean amount of paint per 1-gallon cans and 1 gallon.
b) The p-value obtained = 0.076727 > significance level (0.01), hence, we fail to reject the null hypothesis and conclude that there is no significant evidence to conclude that there there is significant difference in the mean amount of paint per 1-gallon cans and 1 gallon.
That is, the mean amount of paint per 1-gallon cans is not significantly different from 1 gallon.
c) The 99% confidence for the population mean amount of paint per 1-gallon cans is
(0.988, 0.999) in gallons.
d) The result of the 99% confidence interval does not agree with the result of the hypothesis testing performed in (a) because the right amount of paint in 1-gallon cans, 1 gallon, does not lie within this confidence interval obtained.
Step-by-step explanation:
a) This would be answered after solving part (b)
b) For hypothesis testing, the first thing to define is the null and alternative hypothesis.
The null hypothesis plays the devil's advocate and is always about the absence of significant difference between two proportions being compared. It usually contains the signs =, ≤ and ≥ depending on the directions of the test.
While, the alternative hypothesis takes the other side of the hypothesis; that there is indeed a significant difference between two proportions being compared. It usually contains the signs ≠, < and > depending on the directions of the test.
For this question, the null hypothesis is that there is no significant difference in the mean amount of paint per 1-gallon cans and 1 gallon. That is, the mean amount of paint per 1-gallon cans should be 1 gallon.
And the alternative hypothesis is that there is significant difference in the mean amount of paint per 1-gallon cans and 1 gallon. That is, the mean amount of paint per 1-gallon cans is not 1 gallon.
Mathematically,
The null hypothesis is
H₀: μ₀ = 1 gallon
The alternative hypothesis is
Hₐ: μ₀ ≠ 1 gallon
To do this test, we will use the z-distribution because we have information on the population standard deviation.
So, we compute the z-test statistic
z = (x - μ)/σₓ
x = the sample mean = 0.995 gallons
μ₀ = what the amount of paint should be; that is 1 gallon
σₓ = standard error = (σ/√n)
σ = standard deviation = 0.02 gallon
n = sample size = 50
σₓ = (0.02/√50) = 0.0028284271 = 0.00283 gallons.
z = (0.995 - 1) ÷ 0.00283
z = -1.77
checking the tables for the p-value of this z-statistic
p-value (for z = -1.77, at 0.01 significance level, with a two tailed condition) = 0.076727
The interpretation of p-values is that
When the (p-value > significance level), we fail to reject the null hypothesis and when the (p-value < significance level), we reject the null hypothesis and accept the alternative hypothesis.
So, for this question, significance level = 0.01
p-value = 0.076727
0.076727 > 0.01
Hence,
p-value > significance level
So, we fail to reject the null hypothesis and conclude that there is no significant evidence to conclude that there there is significant difference in the mean amount of paint per 1-gallon cans and 1 gallon.
That is, the mean amount of paint per 1-gallon cans is not significantly different from 1 gallon.
c) To compute the 99% confidence interval for population mean amount of paint per 1-gallon paint cans.
Confidence Interval for the population mean is basically an interval of range of values where the true population mean can be found with a certain level of confidence.
Mathematically,
Confidence Interval = (Sample Mean) ± (Margin of error)
Sample Mean = 0.995 gallons
Margin of Error is the width of the confidence interval about the mean.
It is given mathematically as,
Margin of Error = (Critical value) × (standard Error)
Critical value at 99% confidence interval is obtained from the z-tables because we have information on the population standard deviation.
Critical value = 2.58 (as obtained from the z-tables)
Standard error = σₓ = 0.00283 (already calculated in b)
99% Confidence Interval = (Sample Mean) ± [(Critical value) × (standard Error)]
CI = 0.995 ± (2.58 × 0.00283)
CI = 0.995 ± 0.0073014
99% CI = (0.9876986, 0.9993014)
99% Confidence interval = (0.988, 0.999) in gallons.
d) The result of the 99% confidence interval does not agree with the result of the hypothesis testing performed in (a) because the right amount of paint in 1-gallon cans, 1 gallon, does not lie within this confidence interval obtained.
Hope this Helps!!!
