Hypothesis Tests

DNA Pot (c) 2009 - Current

Confidence Interval Method with Large sample size,

σ known

Here, let’s see how to test statistical hypotheses about one population mean when sample size is large. The sample size is considered large when it is greater than or equal to 30. 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

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

This sample is drawn from a population

The probability density function is let say f(x)

The population mean is μ

The population standard deviation is σ

The population variance is σ2

The sample mean is X bar

The sample variance is S2

Then we have three hypothesis we want to test about the population mean, each having one null hypothesis versus an alternative hypothesis.

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,

-zα/2 < z= (X bar - μ)/(σ/√n) < zα/2

-zα/2 < (X bar - μ)/(σ/√n) < zα/2

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

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

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

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

Upper Value: X bar + zα/2 . (σ/√n)

Lower Value: X bar - zα/2 . (σ/√n)

In the interval form, it is written as

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

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.

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