Question

A sample of 44 observations is selected from one population with a population standard deviation of 3.1. The sample mean is 101.0. A sample of 56 observations is selected from a second population with a population standard deviation of 5.0. The sample mean is 99.5. Conduct the following test of hypothesis using the 0.10 significance level.

H0:U1=U2

H1: U1 does not equal U2

a. is this a one or two tailed test?

b.state the decision rule rounded 2 decimals

the decision rule is to reject H0 if z is outside/inside the interval (____,_____)

c. compute the test statistic to two decimals

d. what is your decision regarding H0?

reject/do not reject

e. what is the p value rounded 4 decimals

Answer 1

Step 1: Set up null and alternative hypotheses.

H₀: ?₁ = ?₂             (Note: H₀: ?₁ = ?₂ ? H₀: ?₁ - ?₂ = 0)
H₁: ?₁ ? ?₂             (Note: H₁: ?₁ ? ?₂ ? H₁: ?₁ - ?₂ ? 0)

Based on hypothesis, we can say it is a two tailed test (Answer a)

Step 2: Determine ? (level of significance of hypothesis test).

? = 0.10 (Note: ? = level of significance of hypothesis test = probability of making Type I error.)

Step 3: Calculate test statistic using x?₁, x?₂, ?₁, ?₂, n₁, and n₂; and then

x?₁ for population 1: 101.0
?₁ (population 1 standard deviation) = 3.1  
n₁ (sample size 1) = 44

x?₂ for population 2: 99.5
?₂ (population 2 standard deviation) = 5 n₂ (sample size 2) = 56

              (Note: From Step 1, we have H₀: ?₁ = ?₂ ? H₀: ?₁ - ?₂ = 0; therefore, ?₁ - ?₂ = 0)

Test Statistic z = ((101-99.5)-0)/SQRT(3.1²/44+5²/56)

Test Statistic z = 1.84 (Answer c)

Step 4: Determine P-value

Using the test statistic in Step 3 as a z-score, we will find the left-tailed area and right-tailed area corresponding to this z-score.

Test Statistic = 1.84
Left-tailed area corresponding to z-score of -1.84 = 0.0329 (Obtained using z distribution table)
Right-tailed area corresponding to z-score of 1.84 = 0.0329 (Obtained using z distribution table)
P-value = Left-Tailed Area + Right-Tailed Area = 0.0658

P-value = 0.0658 (Answer e)

Find Critical Values (-z_?/2, z_?/2)

-z_?/2 is the z-score corresponding to the left-tailed area.
z_?/2 is the z-score corresponding to the right-tailed area.

?/2 = 0.1/2 = 0.05

Critical Values are -1.64 and 1.64 (Obtained using z distribution table for ?/2 = 0.05)

The decision rule is to reject H0 if z is outside the interval (-1.64,1.64) Answer (b)

In this case, Test Statistic (1.84) is outside the interval (-1.64,1.64), so we reject null hypothesis H0.

Or we can say since P value (0.0658) < ? (0.10) , so we reject null hypothesis H0 (Answer d)