MATH108 10FA Midterm ch7-11 B
[ answers in web view ] Total points: 70

• Please show all your work! No partial credit will be given for incorrect answers with no work shown.
• Please draw a box around your final answer.
• You are only permitted to use your own calculator and writing implements. Cell phones should be muted and left in your pocket or bag.
• All relevant tables are attached to the back. You may detach them for your reference.
• Assume α = 0.05 everywhere unless indicated otherwise.
• For t-tests on two groups, if the df is not given, you may use the conservative estimate of df = min(n₁, n₂) - 1.

1. Do wives wash hands more frequently than their husbands do? Six couples were asked how many times per day they wash their hands; the results are in the table below.

   							Mean	SD
   Wife:	3	7	8	7	8	9	7	2.098
   Husband:	7	3	3	3	3	5	4	1.673

   
   
   1. What is the population of interest? [1]
      [ Couples ]
   2. Name the variable(s) which need to be measured, indicate their levels of measurement, and whether each is a predictor or outcome variable. [2]
      [ Spouse (wife or husband): categorical/dichotomous (paired).
      Hand washing (times/day): discrete (or continuous). ]
   3. State the null and alternate hypotheses, both in words and in appropriate notation. Which statistical test(s) would be appropriate? [3]
      [ H₀: μ_d ≤ 0 (presuming d = wife - husband; you could also subtract in the other order), wives do not wash their hands more frequently than their husbands.
      H_A: μ_d > 0, wives do wash their hands more frequently than their husbands.
      Paired data: t-test on pairwise differences, or sign test. ]
   4. Run an appropriate parametric test and bracket the p-value. [5]
      
      [ mean diff = 3.00, SD of diffs = 3.464,
      SE_d = 1.414, so t = 2.121.
      df = 5: one-tailed: 0.04 < p < 0.05 (real p = 0.044). ]
   5. State the conclusion from this test, and interpret it in the context of the original research question. [2]
      [ Reject H₀, wives wash their hands more frequently than their own husbands. ]
   6. Using the same data, perform an appropriate non-parametric test and bracket the p-value. [4]
      
      [ Sign test: N₊ = 5, N_- = 1, n_d = 6
      One-tailed: p > .10. ]
   7. State the conclusion from this test, and interpret it in the context of the original research question. [2]
      [ Fail to reject H₀, insufficient evidence to show wives wash their hands more frequently than their own husbands. ]
   8. Which test do you think is more appropriate for this data, the parametric or the non-parametric test? Why? [2]
      [ There is one big outlier -- the first couple. Sign test might be the better option in this case, although overall the sample size is too small to tell. ]
2. Are nurses who work in the ER (emergency room) and nurses who don't work in the ER equally likely to be taking prescription anti-depressants (ATD)? The number of participants in each category is listed in the table below.

   	ATD	No ATD
   ER	33	42
   Non-ER	20	55

   
   
   1. What is the population of interest? [1]
      [ Nurses ]
   2. Name the variable(s) which need to be measured, indicate their levels of measurement, and whether each is a predictor or outcome variable. [2]
      [ ER vs non-ER: categorical/dichotomous.
      ATD: categorical/dichotomous. ]
   3. State the null and alternate hypotheses, both in words and in appropriate notation. Which statistical test(s) would be appropriate? [3]
      [ H₀: P(ATD|ER) = P(ATD|non-ER) (= P(ATD)), or equivalently, P(ER|ATD) = P(ER|no ATD) (= P(ER)):
      whether a nurse is taking prescription anti-depressants is independent of whether that nurse works in the ER or not.
      H_A: P(ATD|ER) ≠ P(ATD|non-ER), or P(ATD|ER) ≠ P(ATD), or P(ER|ATD) ≠ P(ER|no ATD), or P(ER|ATD) ≠ P(ER), etc.
      ER and non-ER nurses are not equally likely to be taking anti-depressants.
      Use the chi-squared test.]
   4. Run the appropriate test and bracket a p-value. [4]
      [ Expected frequencies: 26.5, 48.5, 26.5, 48.5.
      Chi-squared = 4.93, .025 < p < .05 (exact p = 0.026) ]
   5. State the conclusion from this test, and interpret it in the context of the original research question. [2]
      [ Reject H₀, workplace (ER vs. non-ER) and anti-depressant use are not independent. ]
3. Suppose we wish to determine whether the location of birth (in hospital or at home) has an impact on birth weight (in kg).
   1. Name the variable(s) which need to be measured, indicate their levels of measurement, and whether each is a predictor or outcome variable. [2]
      [Birth location: predictor, categorical/dichotomous.
      Birth weight: outcome, continuous]
   2. State the null and alternate hypotheses, both in words and in appropriate notation. Which statistical test(s) would be appropriate? [3]
      [H₀: μ_hosp = μ_home: mean birth weight for both groups is the same; i.e., location does not have an effect on birth weight.
      H_A: μ_hosp ≠ μ_home: mean birth weight differs for the two groups; i.e., location has an effect on weight.
      t-test on independent groups, or Wilcoxon-Mann-Whitney. ]
   3. Data for this experiment are given below. Run an appropriate parametric test and bracket the p-value. [5]
      

      									Mean:	SD:
      Hospital:	3.7	3.2	4.4	4.8	5.1	4.7	2.9	3.2	4.0	0.8519
      Home:	3.6	2.6	3.9	2.7	3.1	3.0	2.8		3.1	0.4830

      
      [ SE₁ = 0.3012, SE₂ = 0.1826, SE = 0.3522.
      mean diff = 0.90, so t = 2.555.
      estimate df = min(n₁, n₂) - 1 = 6
      One-tailed: 0.02 < p < 0.025, so Two-tailed (which is what we want): 0.04 < p < 0.05. (real p = 0.014, with a real df of 11.31). ]
   4. State the conclusion from this test, and interpret it in the context of the original research question. [2]
      [ p < α: reject H₀, location of birth does have an impact on birth weight. ]
   5. Using the same data, perform an appropriate non-parametric test and bracket the p-value. [4]
      
      [ WMW: K₁ = 47, K₂ = 9.
      n = 8, n' = 7 (two-tailed): 0.02 < p < 0.05. ]
   6. State the conclusion from this test, and interpret it in the context of the original research question. [2]
      [ p < α: reject H₀, location of birth does have an impact on birth weight. ]
   7. Which test do you think is more appropriate for this data, the parametric or the non-parametric test? Why? [2]
      [ Parametric should be fine; although the data are more uniformly distributed than normal, there are no obvious outliers. ]
4. Does workplace (defined as "hospital", "clinic", or "community") affect nurses' hourly wage?
   1. What is the population of interest? [1]
      [ Nurses ]
   2. Name the variable(s) which need to be measured, indicate their levels of measurement, and whether each is a predictor or outcome variable. [2]
      [ Workplace: predictor, categorical.
      Wage: outcome, continuous. ]
   3. What is the appropriate parametric statistical test to run? [1]
      [ ANOVA ]
   4. State the null and alternate hypotheses, both in words and in notation. [2]
      [ H₀: μ_h = μ_cl = μ_com,
      i.e., all the nurses have the same wage, regardless of workplace.
      H_A: μ_h ≠ μ_cl, or μ_h ≠ μ_com, or μ_cl ≠ μ_com;
      i.e., at least one group of nurses has a different wage from the other groups. Workplace does have an effect on wage. ]
   5. Data for this experiment are given below. Run an appropriate test and bracket the p-value. [5]
      

      Hospital:	28	30	32
      Clinic:	31	37
      Community:	24	28

      
      [ Grand mean = 30, group means = 30, 34, 26.
      SSb = 64, SSw = 34, dfb = 2, dfw = 4, so MSb = 32, MSw = 8.5.
      F = 3.76, at df = (2,4), so omnibus .10 < p < .15 (actual p = 0.120). ]
   6. State the conclusion from this test, and interpret it in the context of the original research question. [2]
      [ Fail to reject H₀, insufficient evidence to show workplace affects hourly wage. ]
   7. What are the assumptions of the statistical test you performed? Is there evidence to suggest that any of these assumptions have been violated in this dataset? [4]
      [ Parametricity: (1) DV is continuous or discrete (ok)
      (2) Random sample: independent observations, independent groups (ok)
      (3) Normal distribution of DV within each group (awfully small sample size, so can't really tell, but no obvious outliers)
      (4) Equality (homogeneity) of variance (dispersion) amongst groups (also hard to tell, but the min/max spread looks reasonably comparable) ]