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Chapter 11. Hypothesis Test about Population Variance ( Standard Deviation ). Variance. Variance Squared standard deviation Sample variance: s 2 Population variance: σ 2 Measuring the spread (the variability , the dispersion ) of the data around the mean

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### Chapter 11

Hypothesis Test about Population Variance (Standard Deviation)

Variance

- Variance
- Squared standard deviation
- Sample variance: s2
- Population variance: σ2

- Measuring the spread (the variability, the dispersion) of the data around the mean
- A low variance indicates that the data is clustered around the mean (less variation, more consistency)
- A high variance indicates that the data are widely spread out of the mean (more variation, less consistency)

- Squared standard deviation

Variance

- Firm A: Mean= 5%, Standard Deviation (SD)= 3.3%, Variance = (3.3%)2
- Firm B: Mean= 5%, Standard Deviation (SD)= 1.0%, Variance = (1.0%)2

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Hypothesis Tests for Population Variance

- Two hypothesis test procedures for population variance:
- Hypothesis test for a single population variance
- Hypothesis test for two population variances

Hypothesis test for a single population variance

- Example:
- A lumber company has claimed that the standard deviation for the lengths of their 6 foot boards is 0.5 inches or less. To test their claim, a random sample of 17 six-foot boards is selected; and it is determined that the standard deviation of the sample is 0.4. Do the results of the sample support the company's claim?
- In this example, the lumber company claims that the lengths of their 6-foot boards are quite consistent (not much variation in the lengths of their products) because the standard deviation is small (≤ 0.5)

Hypothesis test for a single population variance

- Ho: σ2 = value(σ02)
- Ha: σ2 ≠ value (σ02)

- Ho: σ2 ≥ value (σ02)
- Ha: σ2 < value (σ02)
- Ho: σ2 ≤ value (σ02)
- Ha: σ2 > value (σ02)

Hypothesis test for a single population variance

- Step 1.Formulate H0 and Ha in terms of the population variance (NOT standard deviation)
- Step 2. Compute the sample variance s2 (formula 11.1 on page 450)
Assume that s2 is an estimate for σ2 .

If the sample variance s2 is consistent with Ha, then Ha is supported.

So next, compute the probability that this result is wrong.

Hypothesis test for a single population variance

- Step 3. Compute the chi-square (χ2) by using Formula 11.8 on page 454
- σ02is the number in the hypothesis. DO NOT SQUAREthis number

- Step 4. Compute the p-value by using (1) the χ2, the d.f. (d.f. = n – 1) , and the p-value calculator from χ2 (http://www.danielsoper.com/statcalc/calc11.aspx)
- If Ha has >, use upper-tail p-value
- If Ha has <,use lower-tail p-value
- If Ha has ≠, use two-tailp-value

Hypothesis test for a single population variance

- Step 5. Determineα
- Step 6. If p-value ≤ α, accept the result supporting Ha to be true

Step 4. Compute the p-value

- The calculator only gives you the upper-tail p-value
- To compute the lower-tail p-value:
- lower-tail p-value = 1 – (upper-tail p-value)

- To compute the two-tail p-value:
- Step 1. Compute the upper-tail p-value and the lower-tail p-value
- Step 2. Select the smaller value between those two p-values
- Step 3. Doublethat value for the two-tail p-value

Hypothesis test for a single population variance

- Examples:
- Problem 10 on page 459
- Variance is squared standard deviation
- There is no hypothesis test for standard deviation
- To test a statement/claim about standard deviation, you have to transform the standard deviation into variance and do the hypothesis test for variance
- n = 36 ; s2 = (0.222)2 ; α= 0.05

- Problem 10 on page 459

How to formulate the hypotheses?

- Step 1. Identify what the σ2 (population variance) represents
- Step 2. Find the sentence in the problem that says that the σ2 (population variance) is:
- Greater than (exceed, has increased),
- Less than (fewer than, has been reduced),
- Not equal to (is not, different from, has changed)

- Step 3. Translate the sentence into statistical/mathematical statement:
- σ2 > value,
- σ2 < value,
- σ2 ≠ value

Hypothesis test for two population variance

- This testing procedure is used to test:
- If population one has greatervariance than population two
- If population one has differentvariance from the population two

Hypothesis test for two population variance

- Example:
- The standard deviation of the ages of a sample of 16 executives from the northern states was 8.2 years; while the standard deviation of the ages of a sample of 25 executives from the southern states was 12.8 years.
- Test to see if executives from the northern states are less diverse than the ones from the southern states.

Hypothesis test for two population variance

- Two-Tail Test
- Ho: σ12= σ22
- Ha: σ12≠ σ22

Upper-Tail Test

- Ho: σ12 ≤ σ22
- Ha: σ12 > σ22
Attention:

- Always use the larger sample variance (s2) as σ12
- σ12 is the population variance that represents the larger sample variance (s2)

- Never use lower-tail test
(never use Ha: σ12 < σ22)

Hypothesis test for two population variance

- Step 1.Formulate H0 and Ha in terms of the two population variances
(Remember: Ha: σ12 > σ22 ORHa: σ12 ≠ σ22 , NEVER Ha: σ12 < σ22)

- Step 2. Compute the sample variance s2 (formula 11.1 on page 450)
Assume that s2 is an estimate for σ2 .

If the sample variance s2 is consistent with Ha, then Ha is supported.

So next, compute the probability that this result is wrong.

Hypothesis test for two population variance

- Step 3. Compute Fby using Formula 11.10 on page 461
- Step 4. Compute the p-value by using (1) the F, the d.f. of the two populations(d.f. = n – 1; numerator d.f. is d.f. for the larger variance; denominator d.f. is the d.f. for the smaller variance) , and the p-value calculator from F(Use the P from F calculator in http://www.graphpad.com/quickcalcs/pvalue1.cfm)
- If Ha has > use Upper-tail p-value
- If Ha has ≠ use Two-tail p-value

Hypothesis test for two population variance

- Step 5. Determine α
- Step 6. If p-value ≤ α, accept the result supporting Ha to be true

Step 3. Compute F by using Formula 11.10 on page 461

- Remember:
- Which variance is the s12?
- The larger/greatersamplevariance

- Which variance is the s22?
- The smaller sample variance

- HENCE, F is always > 1.
If F is < 1, you have to switch the two variances

- Which variance is the s12?

Step 4. Compute the p-value

- Which d.f. is the d.f.1(the numerator d.f.)?
- The d.f. computed from the sample that has the larger/greater variance
- d.f.1 is NOT always the larger d.f.

- Which d.f. is the d.f.2(the denominator d.f.)?
- The d.f. computed from the sample that has the smaller variance
- d.f.2 is NOT always the smaller d.f.

Step 4. Compute the p-value

- The calculator only gives you the upper-tail p-value
- To compute the two-tail p-value:
- Step 1. Compute the upper-tail p-value and the (1 – the upper-tail p-value )
- Step 2. Select the smaller value between those two p-values
- Step 3.Double that value for the two-tail p-value

Hypothesis test for two population variance

- Examples:
- Problem 16
- Which variance is the s12?
- Which variance is the s22?
- Which d.f. is the d.f.1 (the numerator d.f.)?
- Which d.f. is the d.f.2 (the denominator d.f.)?

- Problem 18
- Which variance is the s12?
- Which variance is the s22?
- Which d.f. is the d.f.1 (the numerator d.f.)?
- Which d.f. is the d.f.2 (the denominator d.f.)?

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