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Nancy Pfenning
nancyp+@pitt.edu
  

Statistics in a Modern World 800
Solutions to Practice Exam 4

  1. (a) Apparently fathers' mean age is higher than mothers' mean age, because the interval for the difference of the means, (1.23, 4.40), clearly contains only positive numbers.
  2. (c) Mean earnings could be the same, because the intervals (2,4) and (3,5) share a substantial amount of overlap.
  3. (a) Sample mean weight may be biased; a shopper may tend to select lighter, or heavier, berries.
  4.  
    1. (i) z test about a proportion (the word "minority" implies the test's alternative hypothesis is that the population proportion is less than .5)
    2. (ii) z test about a mean (alternative hypothesis is that population mean is less than 98.6)
    3. (iv) chi-squared test (for a relationship between two categorical variables, gender and handedness)
  5. (b) very small P-value means very strong evidence of a relationship
  6. (ii) two-sided alternative makes it more difficult to reject the null hypothesis, because the P-value is larger (twice the one-sided P-value)
  7.  
    1. mean of sample proportions = population proportion = .5
    2. standard error = square root of (.5)(1-.5)/64 = .0625
    3. shape approximately normal because the sample is pretty large (64)
    4. 95% within 2 standard errors of the mean: between .5-2(.0625)=.375 and .5+2(.0625)=.625
    5.  
      1. no: mean of sample proportions is always population proportion (in this case, .5) no matter what the sample size
      2. yes: standard error goes from square root of (.5)(1-.5)/64 to square root of (.5)(1-.5)/4
      3. yes: shape would be less normal because the sample size (4) is so small
  8.  
    1. (v) equal to .15 (proposed population proportion)
    2. (iv) less than .15 (we suspect population proportion is less than what it is nationally)
    3. square root of (.15)(1-.15)/100 = .0357
    4. z = (.08-.15)/.0357 = -1.96
    5. Because the alternative hypothesis says "less than", P-value is probability of z < -1.96 = .025
    6. yes: small P-value means reject null hypothesis and conclude the alternative is true: population proportion is less than .15
    7. (iii) For a cut-off probability of .05, 5% of the time we will reject the null hypothesis, even if it is true.
  9.  
    1. 98.20 plus or minus 2(.6)/(square root of 100) = 98.20 plus or minus .12 = (98.08, 98.32)
    2. yes, because 98.6 is not in the interval
    3. (ii) more subjects yields a narrower interval because the margin of error includes a DIVISION by square root of sample size
    4. (i) is the only correct interpretation. Rule out (ii) and (iii) because the confidence interval is about MEAN temperature, not the temperatures themselves. Rule out (iv) because population mean does NOT vary; it is a fixed parameter. Rule out (v) because the confidence interval is about POPULATION mean, not SAMPLE mean. In fact, we are 100% confident that sample mean is in the interval---we built our interval around it!
  10. (a) a very small sample tends to produce a small z, which results in a large P-value, and failure to reject
  11. (b) a very large sample tends to produce a large z, which results in a small P-value, and strong evidence of a difference which may be unimportant

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