Question

The level of calcium in the blood of healthy young women follows a normal distribution with mean μ = 10.5 milligrams per deciliter and standard deviation σ = 0.25 . To find out if the calcium level changes during pregnancy, a group of 30 healthy pregnant young women is randomly selected and their the calcium level at their first visit for prenatal care is measured.

Specifically, the following hypotheses are tested:

H₀: μ p = 10.5 , H_a: μ p ≠ 10.5 ,

where μ p is the mean calcium level of healthy pregnant young women. Assume the standard deviation of this population, σ p is the same as σ .

Step 1: The mean calcium level of this group of 30 women is 10.43 milligram per deciliter. Find the test statistic. Give your answer to 3 decimal places.

Step 2: Ignore your answer in Step 1. Assume that the test statistic is -1.93. Find the P-value of the test. Give your answer to 4 decimal places.

Step 3: In the above hypothesis testing, we reject the null hypothesis at α = 5 % if the test statistic is:

Answer 1

Solution:

The level of calcium in the blood of healthy young women follows a normal distribution with mean μ = 10.5 milligrams per deciliter and standard deviation σ = 0.25 .

Sample size = n = 30

the following hypotheses are tested:

H₀: μ_p = 10.5 , H_a: μ_p ≠ 10.5

Step 1:

Sample mean = mg

Step 2:

the test statistic is -1.93

Find the P-value of the test.

For two tailed test , P-value is:

P-value = 2* P(Z > z test statistic) if z is positive

P-value = 2* P(Z < z test statistic) if z is negative

Thus

P-value = 2* P(Z < -1.93)

Look in z table for z = -1.9 and 0.03 and find corresponding area.

P( Z < -1.93 ) = 0.0268

Thus

P-value = 2* P(Z < -1.93)

P-value = 2* 0.0268

P-value = 0.0536

Step 3:

In the above hypothesis testing, we reject the null hypothesis at α = 5% if the test statistic is:.......?

Find z critical values:

level of significance =

Since this is two tailed, we find : Area =

Look in z table for area = 0.0250 or its closest area and find z value

Area 0.0250 corresponds to -1.9 and 0.06

thus z critical value = -1.96

Since this is two tailed test, we have two z critical values: ( -1.96 , 1.96)

Thus:

We reject the null hypothesis at α = 5 % if the test statistic is or  if the test statistic is .