Determine the test statistic (z* or t*) and the p-value for each of the followin

Question

Determine the test statistic (z* or t*) and the p-value for each of the following situations and Determine if they would cause the rejection of the null hypothesis if the confidence level was set at 95% in each case. (Hint: be wary of the sample size): a) Ho: mu = 50 mL, Ha: mu not equal to 50 mL, sample mean = 47.3 mL, sample standard deviation = 5, n = 20 b) Ho: mu less than or equal to 8.9 m^3, Ha: mu > 8.9 m^3, sample mean = 10 m^3, s = 3.5, n = 75 c) Ho: mu greater than or equal to 20^o C, Ha: mu < 20^o C , sample mean = 17.1^o C, s = 4.8^o C, n = 12 d) Ho: mu = 380 s, Ha: mu not equal to 380 s, sample mean = 400 s, s = 75, n = 40 e) Ho: mu = 48 units, Ha: mu not equal to 48 units, sample mean = 50 units, s = 9.5, n = 41

Explanation / Answer

Calculate the test statistic and p-value for each sample?
a. H0: = 60 versus H1: _= 60, = .025, ¯x = 63, = 8, n = 16

b. H0: >= 60 versus H1: < 60, = .05, ¯x = 58, = 5, n = 25

c. H0: <= 60 versus H1: > 60, = .05, ¯x = 65, = 8, n = 36

answer-
I will assume that since we are provided with that this is the population standard deviation. I will also assume that the data follows a normal distribution. With these two assumptions I will be able to use a z-test. without these assumptions the z-test would not be valid.

Hypothesis Test for mean:

Assuming you have a large enough sample such that the central limit theorem holds, or you have a sample of any size from a normal population with known population standard deviation, then to test the null hypothesis
H0: or
H0: or
H0: =
Find the test statistic z = (xbar - ) / (sx / (n))

where xbar is the sample average
sx is the sample standard deviation, if you know the population standard deviation, , then replace sx with in the equation for the test statistic.
n is the sample size

The p-value of the test is the area under the normal curve that is in agreement with the alternate hypothesis.

H1: > ; p-value is the area to the right of z
H1: < ; p-value is the area to the left of z
H1: ; p-value is the area in the tails greater than |z|

If the p-value is less than or equal to the significance level , i.e., p-value , then we reject the null hypothesis and conclude the alternate hypothesis is true.

If the p-value is greater than the significance level, i.e., p-value > , the significance level then we fail to reject the null hypothesis and conclude that the null is plausible. Note that we can conclude the alternate is true, but we cannot conclude the null is true, only that it is plausible.


A)

The hypothesis test in this question is:

H0: = 60 vs. H1: 60

The test statistic is:
z = ( 63 - 60 ) / ( 8 / ( 16 ))
z = 1.5

The p-value = P( Z > |z| )
= P( Z < -1.5 ) - P( Z > 1.5 )
= 2 * P( Z < -1.5 )
= 0.1336144

Since the p-value is greater than the significance level of 0.025 we fail to reject the null hypothesis and conclude = 60 is plausible.


B)


The hypothesis test in this question is:

H0: 60 vs. H1: < 60

The test statistic is:
z = ( 58 - 60 ) / ( 5 / ( 25 ))
z = -2

The p-value = P( Z < z )
= P( Z < -2 )
= 0.02275013

Since the p-value is less than the significance level of 0.05 we reject the null hypothesis and conclude the alternate hypothesis < 60 is true.


C)

The hypothesis test in this question is:

H0: 60 vs. H1: > 60

The test statistic is:
z = ( 65 - 60 ) / ( 8 / ( 36 ))
z = 3.75

The p-value = P( Z > z )
= P( Z > 3.75 )
= 8.841729e-05

Since the p-value is less than the significance level of 0.05 we reject the null hypothesis and conclude the alternate hypothesis > 60 is true.

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