An article presents data on dissolved oxygen concentrations in streams below 20

Question

An article presents data on dissolved oxygen concentrations in streams below 20 dams in the TVA system.The sample has an average of 4.25 mg/l. It is known that the population variance is 4.5 (mg/lt)ˆ2.

A) Test the hypothesis H0: Mu=4 versus H1: Mu not equal to (=/) 4 use a alpha of 0.01, what is your conclusion?

B) Compute the power of the test if the true mean dissolved oxygen concentration is as low as 3.

C) What Sample size would be required to detect a true mean dissolved oxygen concentration al low as 2.5 if we wanted the power of the test to be above 0.90?

Explanation / Answer

Here we have to test the hypothesis that,

H0 : mu = 4 Vs H1 : mu not= 4

where mu is population mean.

Assume alpha = level of significance = 0.01

Given that mu = 4

Xbar = sample mean = 4.25

population standard deviation = sigma = 4.5

sample size = n = 20

Here population standard deviation is known then we use one sample z-test.

We can do one sample z-test in MINITAB.

steps :

STAT --> Basic statistics --> 1 sample Z --> Click on summarized data --> ENTER all the values --> Click on perform hypothesis test --> Hypothesized mean : 4 --> Options --> Confidence level : 99.0 --> Alternative : not equal --> ok --> ok

————— 7/2/2017 6:18:16 PM ————————————————————

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One-Sample Z

Test of mu = 4 vs not = 4
The assumed standard deviation = 4.5

N Mean SE Mean 99% CI Z P
20 4.25000 1.00623 (1.65812, 6.84188) 0.25 0.804

Here test statistic Z = 0.25

P-value = 0.804

P-value > alpha

Fail to reject H0 at 0.01 level of significance.

COnclusion : There is sufficient evidence to say that the population mean is 4

B) Compute the power of the test if the true mean dissolved oxygen concentration is as low as 3.

Now we can find power using MINITAB.

steps :

Stat --> Power and sample size --> 1 sample Z --> Input all the values --> Options --> Alternative : not equal --> Significance level : 0.01 --> ok -> ok

1-Sample Z Test

Testing mean = null (versus not = null)
Calculating power for mean = null + difference
Alpha = 0.01 Assumed standard deviation = 4.5

Sample
Difference Size Power
3 20 0.657480

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