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Hypothesis Testing - Statistical Test of Population Mean with known Variance - Theory & Examples

[Home] [Introduction] [Mean (var. known)] [Variance] [Mean (var. unknown)] [Pop. Proportion]

Theory: [Population] [Sample] [Overview]
Examples: [Critical Value] [P-Value] [Type II Error] [Sample size] [Confidence Intervals for Population Mean] [Confidence Intervals for Sample Mean]

Statistical Hypothesis: Testing Mean with known Variance -- Population

Population distribution:

Variance is known, expected value is unknown

Statistical Hypothesis: Testing Mean with known Variance -- Sample

Sample statistics:

sample mean

Sample distribution:

sample distribution of mean

Statistical Hypothesis: Testing Mean with known Variance -- Critical Region

Table overview:

Null Hypotesis	Alternative Hypotesis	Critical Region


Accept H₀	Correct Probability p = 1 - α	Type II Error Probability p = β
Reject H₀	Type I Error Probability p = α	Correct Probability p= 1 - β


Accept H₀
Reject H₀

shaded area = probability of (correctly) accepting the null hypothesis when it is true = 1 - type I error

shaded area = probability of accepting the null hypothesis when it is false = type II error = beta

shaded area = probability of rejecting the null hypothesis when it is true = type I error = alfa

shaded area = probability of (correctly) rejecting the null hypothesis when it is false = 1 - type II error

Statistical Hypothesis: Testing Mean with known Variance -- Example 1: Critical Value (Region)

Free Statistics Software (Calculator)

Problem

problem: find the critical value

Solution

solution: the probability that the null hypothesis is rejected when it is true is equal to alfa. This equation is written in the form of an integral

It is known that if

we introduce an auxiliary variable u

then it follows that

u has a standard normal distribution .

now the alfa probability can be rewritten

the mean is smaller than the critical value

Conclusion

conclusion: do not reject the null hypothesis

Statistical Hypothesis: Testing Mean with known Variance -- Example 2: P-value (probability)

Free Statistics Software (Calculator)

Problem

problem: find the p-value

Solution

solution: we know that the sample mean is normally distributed

the sample mean is standardized

we use the standard normal distribution to compute the p-value .

Conclusion

since the p-value (= 0.22) is larger than alfa (= 0.05) there is no reason to reject the null hypothesis

Statistical Hypothesis: Testing Mean with known Variance -- Example 3: Type II Error

Free Statistics Software (Calculator)

Problem

problem: find the type II error (= beta)

Solution

solution: the probability that the null hypothesis is rejected when it is false is equal to 1 - beta

the critical value is substituted

beta = 0.74 .

Conclusion

conclusion: the type II error (beta) is 74%

Statistical Hypothesis: Testing Mean with known Variance -- Example 4: Sample size

Free Statistics Software (Calculator)

Problem

Solution (method 1)

.

.

.

Conclusion

Solution (method 2)

Conclusion

Statistical Hypothesis: Testing Mean with known Variance -- Example 5: Confidence Intervals for Population Mean

Free Statistics Software (Calculator)

Problem

Two-sided confidence interval

.

Right one-sided confidence interval

Left one-sided confidence interval

Statistical Hypothesis: Testing Mean with known Variance -- Example 6: Confidence Intervals for Sample Mean

Free Statistics Software (Calculator)

Problem

Two-sided confidence interval

.

Right one-sided confidence interval

Left one-sided confidence interval

Mean (var. known)

Mean (var. unknown)

Pop. Proportion

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