• multiple choice:

  • [`X] ... m
  • s/Ön ... standard error SE
  • 2 ... m
  • about 50 times ... would NOT
  • wider

• [`X] = 148,000 and s = 62,000

  • n=25, so use t percentage point in interval (24 degrees of freedom). The formula:

    m = _ X   ±t0.025 * s/Ön

    t_{0.025,df = 24} = 2.06, so

    m = 148,000 ±2.06 * 62,000 / 5,

    and the interval is [122456, 173544].
  • The confidence interval in part a) gives a range of plausible values for the mean of the distribution of home sale prices. Look, instead, at the distribution of sample sale prices: sample mean is $148,000 and sample standard deviation is $62,000. The friend paid $206,000 for a home, this is less than one standard deviation above the sample mean ($210,000), not unusual at all.

• Confidence interval for a difference in population means using unmatched data.

  • sample mean for women, [`X]<sub>W</sub>, is 11; sample mean for men, [`X]_M, is 16. sample variance for women, s²_W=10; sample variance for men, s²_M=21.5. This is an unmatched problem, and the 2 population variances are unknown, use formula 8-20. Degrees of freedom for the t value are, using 8-22, 10-2=8. For a 95% confidence interval t_{0.025,df = 8} = 2.31. Using 8-21,

    s2p = 4*s2W + 4*s2M 4 + 4 = 15.75.

    The confidence interval for the difference in women's and men's salaries, m_W - m_M, is

    mW - mM = _ X   W  - _ X   M  ±t0.025* sp*   æ Ö 1 nW + 1 nM

    where n_W = n_M = 5, so

    mW - mM = -5 ±2.31 * 3.97 * 0.632,

    hence

    mW - mM = -5 ±5.8.

  • This only provides very weak evidence that, on average, women earn less than men. Indeed, the confidence interval contains 0 and hence is consistent with no discrimination. Furthermore, this analysis does not take into account possible explanations for this difference (e.g. experience, training, etc.) and so, while a policy of discrimination would likely result in a difference, a difference is not, in itself, proof of discrimination.

• Confidence interval for a difference in population means using matched data. The data is matched by litter.

  • Use formula 8-12. Start by forming the differences between treatment and control, D = Treatment  -  Control. The sample mean difference, [`D], is 3 and the sample variance, s²_D, of D is 3.56. n = 10, so d.f. = 9 and t_{0.025,df = 9} = 2.26.

    D = _ D   ±t0.025,df = 9 * sd/   __ Ö10   ,

    D = 3 ±2.26 * 0.596,

    D = 3 ±1.35.

  • The experiment provides evidence that an interesting environment leads to greater brain development,as measured by brain weight.

• Confidence interval for a proportion. In a sample of size n tires, a proportion, P = 0.10, failed to meet standards.

  • Expected number of 'successes' is 0.10*10 = 1 < 5, so need to use the graphical method. Find P = 0.10 on the horizontal axis and read off the 95% interval from the vertical axis (n=10):

    0 £ p £ 0.45 .

  • Expected number of 'successes' is 0.10*25 = 2.5 < 5, so need to use the graphical method. Find P = 0.10 on the horizontal axis and read off the 95% interval from the vertical axis (half way between n=20 and n=30):

    0.01 £ p £ 0.30.

  • Expected number of 'successes' is 0.10*50 = 5, so can use either method. Using the graphical method, find P = 0.10 on the horizontal axis and read off the 95% interval from the vertical axis (n=50):

    0.04 £ p £ 0.22.

    Using formula 8.27,

    p = P ±1.96 *   æ Ö P(1-P) n   ,

    p = 0.1 ±1.96 *   æ Ö 0.1(0.9) 50   ,

    p = 0.1 ±0.08.

  • Use formula 8.27,

    p = P ±1.96 *   æ Ö P(1-P) n   ,

    p = 0.1 ±1.96 *   æ Ö 0.1(0.9) 200   ,

    p = 0.1 ±0.04.

• Confidence intervals for differences in proportions. Use formula 8-29 for parts a and b:

  • n_us = n_japan = 300. P_us = 0.40 and P_japan = 0.33

    pus - pjapan = Pus - Pjapan ±1.96 *   æ Ö Pus(1-Pus) nus   +    Pjapan(1-Pjapan) njapan   ,

    pus - pjapan = 0.07 ±1.96 * 0.039,

    pus - pjapan = 0.07 ±0.077,

  • n_us = n_la = 300. P_us = 0.40 and P_la = 0.57

    pus - pla = Pus - Pla ±1.96 *   æ Ö Pus(1-Pus) nus   +    Pla(1-Pla) nla   ,

    pus - pla = -0.17 ±0.079,

---

File translated from T_EX by T_TH, version 2.00.
On 24 Mar 1999, 15:30.