Statistical Inference: Hypothesis Tests and Confidence Intervals, Exams of Biostatistics

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STAT 541 DISCUSSION 7

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1 CIs and Hypothesis Tests

Essentially, hypothesis tests and CIs are equivalent. In particular, if μ 0 is not in the 100(1 − α)% CI for μ, we can reject the null hypothesis of the two-sided test for μ = μ 0 , at level α. One-tests are a little different, because we haven’t talked about one-sided CIs.

2 Comparison of Two Means

1. Paired Samples H 0 : δ = 0 HA : δ 6 = 0

Test statistic: T =

d¯ sd/

√

n If H 0 is correct, T has tn− 1 distribution. Therefore,

p-value = 2P (T > |t|) = 2P (tn− 1 > |t|)

We will reject H 0 while p < α, or while |t| > tn− 1 ,α/ 2.

2. Independent Samples
   • Two samples have the same variance(variance is not known)

H 0 : μ 1 = μ 2 HA : μ 1 6 = μ 2

Test statistic: T =

X¯ 1 − X¯ 2

√

s^2 p(1/n 1 + 1/n 2 )

where s^2 p = (n^1 −1)s

(^21) +(n 2 −1)s (^22) n 1 +n 2 − 2. If^ H^0 is correct, T has^ tn^1 +n^2 −^2 distribution. Therefore

p-value = 2P (T > |t|) = 2P (tn 1 +n 2 − 2 > |t|)

we will reject H 0 while p < α, or while |t| > tn 1 +n 2 − 2 ,α/ 2.

• Two samples have unequal variance

H 0 : μ 1 = μ 2 HA : μ 1 6 = μ 2

Test statistic: T =

X¯ 1 − X¯ 2

√

s^21 /n 1 + s^22 /n 2

while H 0 is correct, T has an approximate t distribution with the degrees of freedom v, where

v =

(s^21 /n 1 + s^22 /n 2 )^2 (s^21 /n 1 )^2 /(n 1 − 1) + (s^22 /n 2 )^2 /(n 2 − 1)

the value of v is then rounded down to the nearest integer. Therefore

p-value = 2P (T > |t|) = 2P (tv > |t|)

We will reject H 0 while p < α, or while |t| > tv,α/ 2.

3 Examples

1 Typically having a small number of degrees of freedom hurts us. In particular, it makes it more difficult to reject the null hypothesis. So, why would we want a paired t-test, which only gives us n − 1 dof, when the unpaired test would give us twice as many dof? 2 Suppose that you are interested in determining whether exposure to the organochlo- rine DDT, which has been used extensively as an insecticide for many years, is as- sociated with breast cancer in women. As part of a study that investigated this issue, blood was drawn from a sample of women diagnosed with breast cancer over a six-year period and from a sample of healthy control subjects matched to the can- cer patients on age, menopausal status, and date of blood donation. Each woman’s blood level of DDE- an important byproduct of DDT in the human body-was mea- sured, and the difference in levels for each patient and her matched control was calculated. A sample of 171 such differences has mean d¯ = 2.7 and standard devia- tion sd = 15. 9 ng/ml. (a) Test the null hypothesis that the mean blood levels of DDE are identical for women with breast cancer and for healthy control subjects. What do you con- clude? α = 0. 05 (b) Would you expect a 95% confidence interval for the true difference in population mean DDE levels to contain the value 0? 3 A study was conducted to determine whether an expectant mother’s cigarette smok- ing has any effect on the bone mineral content of her otherwise healthy child. A sample of 77 newborn whose mothers smoked during pregnancy has mean bone min- eral content ¯x 1 = 0. 098 g/cm and standard deviation s 1 = 0. 026 g/cm; a sample of 161 infants whose mothers did not smoke has mean ¯x 2 = 0. 095 g/cm and standard deviation s 2 = 0. 025 g/cm. Assume that the underlying population variances are equal, test for a significant difference of the bone mineral content of the two groups with α = 0.05. 4 Tennis elbow is thought to be aggravated by the impact experienced when hitting the ball. The article ”Forces on the Hand in the Tennis One-Handed Backhand” (Intl. J. of Sport Biomechanics, 1991: 282-292) reported the force (N) on the hand just after impact on a one-handed backhand drive for six advanced players and for eight intermediate players. The advanced players have a mean force of 40.3 and standard deviation 11.3. And the intermediate players have a mean force of 21. and standard deviation 8.3. Test for a significant difference in mean force between two groups of players with level of significance 0.05.