Hypothesis (please answer each problem, then I will rate the answer) According t
ID: 3206389 • Letter: H
Question
Hypothesis (please answer each problem, then I will rate the answer)
According to the American Medical Association, approximately 50% of all U. S. physicians whose ages are under 35 are known to be women. Four samples of 15 physicians were interviewed each. Let X_i be the number of women that were found in the i^th random sample of 15 physicians whose ages are less than 35. The number of women in each of the four samples were counted and it was found that bar x = 5 and .s^2 = 2/3. What is the 95% confidence interval about mu? Continuation of problem 3. Given that bar x = 5 and s^2 = 2/3 from the 4 samples of 15 physicians each, test the hypothesis that mu = 7.5 against the alternative that mu notequalto 7.5 at the .05 level of significance.Explanation / Answer
Solution
Back-up Theory
If X ~ N(µ, 2) and X bar is the sample average and s is the sample standard deviation based on n observations, then
i) X bar ~ N(µ, 2/n)
ii) (n){(X bar - µ)/} ~ N(0, 1), if is known
iiI) (n){(X bar - µ)/s} ~ tn – 1(i.e., t-distribution with (n- 1) degrees of freedom, if is unknown
iv) 100x(1 – )% confidence interval for µ is given by: X bar ± (Z/2.x /n) or X bar ± (tn – 1,/2.x s/n), depending whether is known or is unknown respectively, where Z/2 and tn – 1,/2 are upper /2 percent points of Standard Normal and t-distribution with (n - 1) degrees of freedom respectively, both of which can be read off from Standard Statistical Tables.
Now, to work out the problem,
Given, n = 4, X bar = 5 and s = (2/3) and from Standard Statistical Tables t3, 0.025 = 3.182,
95% Confidence Interval for µ is: 5 ± (3.182 x 0.8165/2) = 5 ± 1.2991
= (3.7, 6.3) ANSWER for Q3
Q4
H0: µ = 7.5 vs HA: µ 7.5
Test Statistic: t = (n){(X bar - µ)/s} = 2(2.5)/0.8165 = 6.124.
The test statistic has a t-distribution with (n - 1) degrees of freedom and H0 is rejected at % level of significance if calculated value of t > upper /2 percent point of t-distribution with (n - 1) degrees of freedom.
Since 6.123 > t3, 0.025 = 3.182, H0 is rejected. ANSWER
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