Comments on Hypothesis Testing a) The null hypothesis is usually the status quo, what you suspect has changed. b) The alternative hypothesis is what you suspect to be different, three choices (less, not equal, greater). c) NEVER use sample data to influence choice of null hypothesis or alternative hypothesis. d) A high P-value means "cannot reject" the null hypothesis. e) A low P-value means "reject" the null hypothesis.
Problem #1: A university bookstore manager is interested in determining if there is convincing evidence that the proportion of students at the university who purchase one or more of their textbooks online is less than a reported national figure of 0.20. What hypothesis should the bookstore manager test? Tell me H0 and Ha.
The manager of a large hotel must decide whether to hire additional front desk staff. He has decided to hire more staff if there is evidence that the average time customers must wait in line before being assisted with check-in is greater than 3 minutes. Tell me H0 and Ha.
Problem #3: The principal at a large high school will implement a proposed after-school tutoring program if there is evidence that the proportion of students at the school who would take advantage of such a program is greater than 0.10. Tell me H0 and Ha.
Problem #4: If the conclusion in a hypothesis test is fail to reject H0, which of the following is an appropriate conclusion? a) There is convincing evidence that the null hypothesis is true. b) There is convincing evidence that the null hypothesis is false. c) There is not convincing evidence that the null hypothesis is false.
Problem #5: Which of the following is approximately equal to the P-value for a test in which the hypotheses are H0: p = 0.5 versus Ha: p ≠ 0.5 and for which the value of the test statistic is z = -2? a) 0.025 b) 0.05 c) 0.95 d) ).975
Problem #7: To determine whether to produce a new design of a baseball cap with a team logo printed on the back, one hundred men who were wearing a baseball cap were selected at random from those attending a major league baseball game. On these men, 27 wore the hat with the bill facing backward. This data was used to test the hypotheses H0: p = 0.33 versus Ha: p < 0.33 where p is the proportion of baseball cap wearing men at the game who wear the cap with the bill facing backward. A significance level of .05 was used, and the P-value for this test was 0.101. If the new hat design will be produced unless there is convincing evidence that fewer than one third wear the hat backward, what is an appropriate decision based on this P-value?
a) Produce the cap, because there is convincing evidence that less than one third wear the hat backward. b) Don't produce the cap, because there is convincing evidence that less than one third wear the hat backward. c) Produce the cap, because there is not convincing evidence that less than one third wear the hat backward. d) Don't produce the cap, because there is not convincing evidence that less than one third wear the
Text Problem, page 520 Do home baseball teams really have an advantage? Do they win more at home? Statistically: do they win significantly more than they should? H0: p = 0.50
Ha: p > 0.50
The data from 2006 MLB: 2429 games 1327 won by home team, 54.63% Independent: Random: 10% condition: np, nq > 10
probably not, but pretty close one full year, a reasonable sample 2006 year < all games over many years easily verified
The Calculations: This is a one-tail test, to the right! P-value = Pr(p > 0.5463)
convert to z-score then find area
z = 4.56; Pr(z > 4.56) = normalcdf(4.56, 100) = 0.0000026 If the true proportion of home team wins were 0.50 then the observed value of 0.5463 would occur less than 3 in a million samples. With such a small P-value we reject the null hypothesis and conclude that the true proportion is not 0.50 and that there is a home field advantage.
Text #29, page 530 A company is criticized because only 13 of 43 people in executive-level positions are women. The company explains that although this proportion is lower than it might wish, it's not surprising given only 40% of all its employees are women. What do you think? ... think statistically! Test an appropriate hypothesis and state your conclusion.
Data only for this company, cannot generalize to others. H0: p = 0.40 Ha: p < 0.40 one tail test, to left z = -1.31 P-value = 0.0955, about 10% The p-value is not small, it is large enough to not reject the null hypothesis. These data do not show that the proportion of women executives is less than the 40% of women in the company in general.
Text #22, page 529 According to the Association of American Medical Colleges, only 46% of medical school applicants were admitted to a medical school in the fall of 2006. Upon hearing this, the trustees of Striving College expressed concern that only 77 of the 180 students in their class of 2006 who applied to medical school were admitted. The college president assured the trustees that this was just the kind of year-to-year fluctuation in fortunes that is to be expected and that, in fact, the school's success rate was consistent with the national average. Who is right?
H0: p = 0.46 Ha: p < 0.46
a one-tail test
Check independence, random, 10%, np&nq >10 z = -0.87 P-value = 0.19 = 19% The high P-value indicates that this is not an unusual result; it may be just the year-to-year variation, as the president says.