Inference about Two Means: Dependent Samples Section 10'1

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Inference about Two MeansDependent Samples
Section 10.1

• Alan Craig
• 770-274-5242
• acraig_at_gpc.edu

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Objectives 10.1

• Distinguish between independent and dependent
  sampling
• Test claims made regarding matched-pairs data
• Construct confidence intervals about the
  population mean difference of matched-pairs data

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Independent versus Dependent Sampling

• A sampling method is independent when the
  individuals selected for one sample do no dictate
  which individuals are to be in a second sample.
• A sampling method is dependent when the
  individuals selected for one sample are used to
  determine the individuals to be in a second
  sample.
• Matched-pairs sampling is a dependent sampling
  method.
• Determine whether sampling is dependent or
  independent in 2, 4, 6 on p. 442

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Matched-Pairs

• Do married couples have similar IQs?
• We select a random sample of married couples and
  determine the IQs of husbands and wives
• If we averaged IQs for all husbands and then for
  all wives and compared the two averages, would
  that answer our question?

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Matched-Pairs

• No. If we did that, we would have the average of
  a bunch of women and a bunch of men.
• However, we could compute the difference in each
  couples IQs. (Wife1 Husband1)
• We could then test whether the mean of these
  differences is significantly different from 0.

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Testing a Claim about the Difference of Two Means
Using a Matched-Pairs Design

• Requirements
• Simple random sampling
• Sample data are matched-pairs
• Differences are (approximately) normally
  distributed or n 30

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Testing a Claim about the Difference of Two Means
Using a Matched-Pairs Design

• Step 1 State the hypotheses

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Testing a Claim about the Difference of Two Means
Using a Matched-Pairs Design

• Step 2 Choose a and find the critical t-value
  with n 1 degrees of freedom

Left-Tailed Two-Tailed
Right-Tailed
Critical Region Reject H0
Critical Region Reject H0
Critical Region Reject H0
Critical Region Reject H0
- ta
ta
- ta/2
ta/2
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Testing a Claim about the Difference of Two Means
Using a Matched-Pairs Design

• Step 3 Compute the test statisticapproximately
  follows a t-distribution with n 1 df
• are the mean and standard
  deviation of the differenced data.

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Testing a Claim about the Difference of Two Means
Using a Matched-Pairs Design

• Step 4 Compare the critical value with the test
  statistic
• Left-Tailed Two-Tailed Right-Tailed
• t lt -ta t lt -ta/2 or t gt ta/2
  t gt ta
• Reject H0 Reject H0 Reject H0
• Step 5 State the conclusion.

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Example 16 (a), p. 445

• A test prep company claims its SAT prep course
  improves SAT math scores. Company administers SAT
  to 12 randomly selected students and determines
  their scores. The same students then take the
  course. They then retake the SAT. Test the claim
  at the a 0.10 level of significance.
• Before
• 436 431 270 463 528 377 397 413 525 323 413 292
• After
• 443 429 287 501 522 380 402 450 548 349 403 303
• Enter Before in L1 and After in L2

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Example 16 (a), p. 445

• First calculate the differences
• On the calculator, first L2 L1?L3
• Why do we subtract L1 from L2 in this case and
  not vice versa?

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Example 16 (a), p. 445

• Verify Requirements Because n lt 30, we need to
  ensure the data is approximately normal. We have
  the differences in L3, so we do a normal
  probability plot and modified boxplot.
• The normal probability plot is more or less
  linear.

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Example 16 (a), p. 445

• Verify Requirements Because n lt 30, we need to
  ensure the data is approximately normal. We have
  the differences in L3, so we do a normal
  probability plot and modified boxplot.
• The modified boxplot shows no outliers.
• We can proceed.

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Example 16 (a), p. 445

• Step 1 State hypotheses
• Step 2 a 0.10 level of significance.
• df 11, right-tailed test, so t0.10 1.363

Critical Region Reject H0
ta
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Example 16 (a), p. 445

• Step 3 Compute test statistic
• First find the mean and standard deviation of the
  differenced data.
• On the calculator, 1-Var Stats L3
• md 12.417
• sd 15.957

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Example 16 (a), p. 445

• Step 3 Compute test statistic (continued)
• md 12.417 sd 15.957
• Step 4 Compare critical value to test statistic
• t 2.696 gt 1.363 t0.10
• Step 5 We reject H0. There is sufficient
  evidence to show that the SAT prep course
  improves SAT math scores.

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Example 16 (a), p. 445

• If the level of significance is a 0.05, do we
  reject the null hypothesis?
• (test statistic t 2.696 with df 11)

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Example 16 (a), p. 445

• Use calculator to compare critical value to test
  statistic. Assume difference data is in L3 as
  ours is.
• Use STATS?TESTS?2 T-Test
• Type m0 0, List L3, Freq 1, m gtm0
• Calculate Draw

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Confidence Interval for Matched-Pairs Data

• CI Point Estimate Margin of Error
• (1 a)100 confidence interval for md
• ta/2 has n-1 df, are the mean and
  standard deviation of the differenced data.

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Example 16 (b), p. 445

• Construct a 95 confidence interval about the
  population mean difference. Interpret your
  result.
• Step 1 Compute the differenced data and verify
  approximately normal.
• Step 2 Compute
• We have already done these two steps in part (a).
• In particular, we found

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Example 16 (b), p. 445

• Construct a 95 confidence interval about the
  population mean difference. Interpret your
  result.
• Step 3 Determine the critical value ta/2 with
  a 0.05 and n-1 degrees of freedom.
• Using Table III with a 0.05 and 12 1 11 df,
  we find ta/2 t0.025 2.201

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Example 16 (b), p. 445

• Construct a 95 confidence interval about the
  population mean difference. Interpret your
  result.
• Step 4 Compute the confidence interval.
• We are 95 confident that the true mean
  difference in SAT Math scores is between 2.28 and
  22.56 points. That is, we are 95 confident that
  the SAT Prep Course improves Math scores
  somewhere between 2.28 and 22.56 points.

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Example 16 (b), p. 445

• Construct a 95 confidence interval about the
  population mean difference. Interpret your
  result.
• Compute the confidence interval using calculator.
• Use STATS?TESTS?8 TInterval
• Type Inpt Data, List L3, Freq 1, C-Level .95

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Questions

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