SOLUTION: A consumer group claims that the average annual consumption of high fructose corn syrup by a person in the U.S. is 48.8 pounds. You believe it is higher. You take a simple random s

Question 1197695: A consumer group claims that the average annual consumption of high fructose corn syrup by a person in the U.S. is 48.8 pounds. You believe it is higher. You take a simple random sample of 34 people in the U.S. and find an average of 53.8 pounds with a standard deviation of 6 pounds. Test at 10% significance
We can work this problem because:
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H
0
:
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H
A
:
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Test Statistic:
P-value:
Did something significant happen?
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Select the Decision Rule:
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There
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enough evidence to conclude
Found 2 solutions by ewatrrr, math_tutor2020:
Answer by ewatrrr(24785) (Show Source): You can put this solution on YOUR website!

Ho: µ = 48.8 Ha: µ > 48.8 n = 34, x̄ = 53.8 s = 6 As population sd is not known, used a t-test vs a z- test (either is fine n >30) 0.10 significance level, Critical value = 1.3077 (df 33) = 4.859 4.859 > 1.3077 Reject Ho P(z ≤ 4.859, df 33) = .00001 < .05 Reject Ho If Using a Standard z-Distribution: 4.859 > 1.282 Reject Ho P(z ≤ 4.859) = .0000006 < .05 Reject Ho

Answer by math_tutor2020(3817) (Show Source): You can put this solution on YOUR website!

n = 34 = sample size
xbar = 53.8 = sample mean
s = 6 = sample standard deviation

mu = population mean
sigma = population standard deviation
Sigma is unknown, which is a common occurrence in stats.
Despite sigma being unknown, we can use the Z distribution because n > 30.
The T distribution is approximately fairly close to the standard Z distribution for these large values of n.

If sigma were unknown and n < 30, then we'd have to use the T distribution.

Hypothesis:
: mu = 48.8
: mu > 48.8
The claim is in the alternative hypothesis.
This gives us a right tailed test.

Computing the test statistic:
z = (xbar - mu)/(s/sqrt(n))
z = (53.8 - 48.8)/(6/sqrt(34))
z = 4.85912657903776
z = 4.86

Use a Z calculator to find that
P(Z > 4.86) = 0.000000587
which is the approximate p-value.
I don't know how the tutor @ewatrrr got P(z ≤ 4.859) = .000014 as that is not correct.

This p-value 0.000000587 is very small.
Depending how you round this, the p-value is effectively 0.

Whenever the p-value is smaller than alpha, we reject the null.
We conclude that mu > 48.8 appears to be the case.

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If you wanted to go the critical value route, then at 10% significance, the critical value is roughly z = 1.28 since P(Z > 1.28) = 0.10 approximately

The test statistic (z = 4.86) is to the right of the critical value (z = 1.28), which places the test statistic in the rejection region.
This is another way to see why we reject the null in favor of the alternative hypothesis.

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