chapter 11 l.
Download
Skip this Video
Download Presentation
Chapter 11

Loading in 2 Seconds...

play fullscreen
1 / 23

Chapter 11 - PowerPoint PPT Presentation


  • 1315 Views
  • Uploaded on

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

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'Chapter 11' - kevina


Download Now An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
chapter 11

Chapter 11

Hypothesis Test about Population Variance (Standard Deviation)

variance
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)
variance3
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
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
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 variance6
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 variance7
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 variance8
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 variance9
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
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 variance11
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
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
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 variance14
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 variance15
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)

slide16

!

For TWO population variance,

NEVER use

Ha: σ12 < σ22

hypothesis test for two population variance17
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 variance18
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 variance19
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
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 value21
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 value22
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 variance23
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.)?