Use a t-distribution and the given matched pair sample results to complete the test of the given hypotheses. Assume the results
1 answer:
Answer:
Step-by-step explanation:
Given that we can Assume the results come from random samples, and if the sample sizes are small, assume the underlying distribution of the differences is relatively normal. Assume that differences are computed using d=x1-x2. Test H0 : μd=0 vs Ha : μd≠0
Here test statistic = Mean difference ./std error
= Mean difference/(std dev/(sqrt n))
=
df =24
p value = 0.000121
Since p < alpha, we reject null hypothesis.
The two means are significantly different.
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Part A
Given info:
- xbar = sample mean = 4.90 ppm
- s = sample standard deviation = 1.12 ppm
- n = 23 = sample size
Because n > 30 is not true, and we don't know the population standard deviation (sigma), this means we must use a T distribution.
The degrees of freedom here are n-1 = 23-1 = 22.
At 90% confidence and the degrees of freedom mentioned, the t critical value is roughly t = 1.717
Use a T distribution table or calculator to determine this. If you don't have a calculator for the task, then you can search out "inverse T calculator" and there are tons of free options to pick from.
The margin of error E is
E = t*s/sqrt(n)
E = 1.717*1.12/sqrt(23)
E = 0.400982
This is approximate and accurate to 6 decimal places.
The confidence interval is going to be xbar plus or minus that E value
L = lower bound = xbar - E = 4.90 - 0.400982 = 4.499018 = 4.50
U = upper bound = xbar + E = 4.90 + 0.400982 = 5.300982 = 5.30
The confidence interval in the format of (L, U) is (4.50, 5.30)
You could also express it as the format L < mu < U and it would be 4.50 < mu < 5.30; however, I'll stick to the first method.
<h3>Answer: (4.50, 5.30)</h3>=====================================================
Part B
Since we know sigma = 2.6 is the population standard deviation, we can use a Z distribution now.
At 90% confidence, the z critical value is roughly 1.645; use a table or calculator to determine this.
Round this up to the nearest integer to get 20326. For min sample size problems, <u>always</u> round up.
<h3>Answer: 20326</h3>