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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

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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)


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

0

0


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


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)


!

For TWO population variance,

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


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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