Hypothesis Tests

DNA Pot (c) 2009 - Current

Confidence Interval Method with Large sample size,

σ known

Example A random sample of 36 pieces of iron wire produced in a plant of wire manufacturing company yields the mean tensile strength of sample mean X bar = 950 psi. Suppose that the population of tensile strengths of all iron wires produced in that plant are distributed with mean μ and standard deviation σ = 120 psi, test a statistical hypothesis

Ho : μ = 980

(1) Ho : μ < 980

(2) Ho : μ > 980

(3) Ho : μ ≠ 980

at the α = 0.01 level of significance. The population is assumed normal. Test at α = 0.05, using confidence interval for α with confidence coefficient 1 - α, i.e., 95%.

Suppose we have a random sample of    X1, X2, X3, ..., Xn

we have a random sample of 36 pieces of iron wire

the sample size is clearly n = 36, and in this case it is large (n  ⩾ 30)

This sample is drawn from a population

The population mean is μ

The population standard deviation is σ = 120 psi

The sample mean is X bar = 950 psi

Population mean stated in the null hypothesis μ0 = 980 psi

Then we have a hypothesis we want to test about the population mean using a confidence interval, Ho : μ ≠ μ0

And, the level of significance is α = 0.05

Step 1

State the null hypothesis

Ho : μ = μ0

Step 2

State the alternative hypothesis

Ho : μ ≠ μ0

Step 3

Assign an appropriate value of level of significance α

Common Values: 0.01 (1%), 0.05 (5%), and 0.10 (10%)

α = 0.05 is given in this case

Step 4

Determine a suitable test statistic

The parameter under investigation is used as test statistic

The test statistic is

z = (X bar - μ)/(σ/√n)  for  μ

This test statistic is used to test hypothesis Ho : μ ≠ μ0

Step 5

Determine the probability distribution in the test statistic

Since, the sample is large (n ⩾ 30), according to the central limit theorem the test statistic “ z= (X bar - μ)/(σ/√n)  for  μ” is distributed as standard normal (mean 0 and standard deviation 1).

Step 6

Find the upper and lower values of the confidence interval

Confidence interval for α is with confidence coefficient 1 - α, i.e., 95%.

Ho : μ ≠ μ0

z= (X bar - μ)/(σ/√n)  for  μ

The upper and lower values of confidence interval can be found

X bar - zα/2 . (σ/√n) < μ < X bar + zα/2 . (σ/√n)

Upper Value: X bar + zα/2 . (σ/√n) = 950 + [1.96 * (120/√36)] =989

Lower Value: X bar - zα/2 . (σ/√n) = 950 - [1.96 * (120/√36)] = 911

In the interval form, it is written as

(911, 989)

Step 7

Make decision

Determine whether the null hypothesis value falls in this confidence interval. If it falls in between upper and lower values, do not reject the null hypothesis. If it is less than the lower value or greater than the upper value of the confidence interval, reject the null hypothesis value.

911 < 980 < 989

Since μ = μ0 = 980 for null hypothesis and 980 is in between 911 and 989, therefore we do not reject null hypothesis.