Answer:
a)
: Do not reject the null hypothesis.
: Reject the null hypothesis.
b) ![z = 2.81](https://tex.z-dn.net/?f=z%20%3D%202.81)
c) Reject.
d) The p-value is 0.005.
Step-by-step explanation:
Before testing the hypothesis, we need to understand the central limit theorem and the subtraction of normal variables.
Central Limit Theorem
The Central Limit Theorem establishes that, for a normally distributed random variable X, with mean
and standard deviation
, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean
and standard deviation
.
For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.
Subtraction between normal variables:
When two normal variables are subtracted, the mean is the difference of the means, while the standard deviation is the square root of the sum of the variances.
Population 1:
Sample of 42, standard deviation of 3.3, mean of 101, so:
![\mu_1 = 101](https://tex.z-dn.net/?f=%5Cmu_1%20%3D%20101)
![s_1 = \frac{3.3}{\sqrt{42}} = 0.51](https://tex.z-dn.net/?f=s_1%20%3D%20%5Cfrac%7B3.3%7D%7B%5Csqrt%7B42%7D%7D%20%3D%200.51)
Population 2:
Sample of 53, standard deviation of 3.6, mean of 99, so:
![\mu_2 = 99](https://tex.z-dn.net/?f=%5Cmu_2%20%3D%2099)
![s_2 = \frac{3.6}{\sqrt{53}} = 0.495](https://tex.z-dn.net/?f=s_2%20%3D%20%5Cfrac%7B3.6%7D%7B%5Csqrt%7B53%7D%7D%20%3D%200.495)
H0 : μ1 = μ2
Can also be written as:
![H_0: \mu_1 - \mu_2 = 0](https://tex.z-dn.net/?f=H_0%3A%20%5Cmu_1%20-%20%5Cmu_2%20%3D%200)
H1 : μ1 ≠ μ2
Can also be written as:
![H_1: \mu_1 - \mu_2 \neq 0](https://tex.z-dn.net/?f=H_1%3A%20%5Cmu_1%20-%20%5Cmu_2%20%5Cneq%200)
The test statistic is:
In which X is the sample mean,
is the value tested at the null hypothesis, and s is the standard error
.
a. State the decision rule.
0.04 significance level.
Two-tailed test(test if the means are different), so between the 0 + (4/2) = 2nd and the 100 - (4/2) = 98th percentile of the z-distribution, and looking at the z-table, we get that:
: Do not reject the null hypothesis.
: Reject the null hypothesis.
b. Compute the value of the test statistic.
0 is tested at the null hypothesis:
This means that ![\mu = 0](https://tex.z-dn.net/?f=%5Cmu%20%3D%200)
From the samples:
![X = \mu_1 - \mu_2 = 101 - 99 = 2](https://tex.z-dn.net/?f=X%20%3D%20%5Cmu_1%20-%20%5Cmu_2%20%3D%20101%20-%2099%20%3D%202)
![s = \sqrt{s_1^2 + s_2^2} = \sqrt{0.51^2 + 0.495^2} = 0.71](https://tex.z-dn.net/?f=s%20%3D%20%5Csqrt%7Bs_1%5E2%20%2B%20s_2%5E2%7D%20%3D%20%5Csqrt%7B0.51%5E2%20%2B%200.495%5E2%7D%20%3D%200.71)
Value of the test statistic:
![z = \frac{X - \mu}{s}](https://tex.z-dn.net/?f=z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7Bs%7D)
![z = \frac{2 - 0}{0.71}](https://tex.z-dn.net/?f=z%20%3D%20%5Cfrac%7B2%20-%200%7D%7B0.71%7D)
![z = 2.81](https://tex.z-dn.net/?f=z%20%3D%202.81)
c. What is your decision regarding H0?
, which means that the decision is to reject the null hypothesis.
d. What is the p-value?
Probability that the means differ by at least 2, either plus or minus, which is P(|z| > 2.81), which is 2 multiplied by the p-value of z = -2.81.
Looking at the z-table, z = -2.81 has a p-value of 0.0025.
2*0.0025 = 0.005
The p-value is 0.005.