Question

A sample of 42 observations is selected from one population with a population standard deviation of 3.3. The sample mean is 101.0. A sample of 53 observations is selected from a second population with a population standard deviation of 3.6. The sample mean is 99.0. Conduct the following test of hypothesis using the 0.04 significance level.
H0 : μ1 = μ2
H1 : μ1 ≠ μ2
a. State the decision rule.
b. Compute the value of the test statistic.
c. What is your decision regarding H0?
d. What is the p-value?

Answer 1

Answer:

a)

$|z| < 2.054$: Do not reject the null hypothesis.

$|z| > 2.054$: Reject the null hypothesis.

b) $z = 2.81$

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 $\mu$ and standard deviation $\sigma$, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean $\mu$ and standard deviation $s = \frac{\sigma}{\sqrt{n}}$.

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:

$\mu_1 = 101$

$s_1 = \frac{3.3}{\sqrt{42}} = 0.51$

Population 2:

Sample of 53, standard deviation of 3.6, mean of 99, so:

$\mu_2 = 99$

$s_2 = \frac{3.6}{\sqrt{53}} = 0.495$

H0 : μ1 = μ2

Can also be written as:

$H_0: \mu_1 - \mu_2 = 0$

H1 : μ1 ≠ μ2

Can also be written as:

$H_1: \mu_1 - \mu_2 \neq 0$

The test statistic is:

$z = \frac{X - \mu}{s}$

In which X is the sample mean, $\mu$ 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:

$|z| < 2.054$: Do not reject the null hypothesis.

$|z| > 2.054$: Reject the null hypothesis.

b. Compute the value of the test statistic.

0 is tested at the null hypothesis:

This means that $\mu = 0$

From the samples:

$X = \mu_1 - \mu_2 = 101 - 99 = 2$

$s = \sqrt{s_1^2 + s_2^2} = \sqrt{0.51^2 + 0.495^2} = 0.71$

Value of the test statistic:

$z = \frac{X - \mu}{s}$

$z = \frac{2 - 0}{0.71}$

$z = 2.81$

c. What is your decision regarding H0?

$|z| = 2.81 > 2.054$, 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.