For several years, aspirin-based products lost market share to substitute analge

Question

For several years, aspirin-based products lost market share to substitute analgesics such as ibuprofen and acetaminophen. As a result, large companies such as Bayer, which had previously produced only aspirin-based products, began marketing acetaminophen-based pain relievers. The following table gives the proportions of the individuals who preferred each of the three major pain relievers for the years 1986 and 1991.

Assume that these results were obtained on the basis of two independent samples, each of size 1000.

a.) Does statistically significant evidence exist to suggest that the proportions of aspirin users differ for 1986 and 1991? Use a 0.05 level test.

c.) Are the tests performed in (a) and (b) related? Explain.

Pain Reliever 1986 1991 Aspirin 45% 34% Acetaminophen 41% 41% Ibuprofen 14% 26% a 0,05

Explanation / Answer

(a)

Null HYpothesis : H0 : There is no difference between proportion of aspirin users in 1986 and 1991 p1986 = p1991

Alternative Hypothesis : Ha : There is significant difference between proportion of aspirin users in 1986 and 1991 p1986 p1991  

Here p1986 = 0.45 and p1991 = 0.34

Test statistic

pooled estimate p = (1000 * 0.45 + 1000 * 0.34)/ 2000 = 0.395

standard error of population distribution SE0 = sqrt [ p(1-p) (1/n1 + 1/n2 )] = sqrt [0.395 * 0.605 * 2/1000]

= 0.02186

Z = (p1986 -  p1991 )/ se0 = (0.45 - 0.34)/ 0.02186 = 5.032

so for alpha = 0.05 and non - directional Z - test

Zcritical = 1.96

so here Z > Zcritical , so we can reject the null hypothesis and conclude that proportions of aspirin users differs in year 1986 and 1991/

(b)

Null HYpothesis : H0 : There is no difference in ibuprofen market share in 1986 and 1991 p1986 = p1991

Alternative Hypothesis : Ha : There is significant increase in ibuprofen market share in 1991 from the year 1986. p1986 <  p1991  

Here p1986 = 0.14 and p1991 = 0.26

Test statistic

pooled estimate p = (1000 * 0.14 + 1000 * 0.26)/ 2000 = 0.2

standard error of population distribution SE0 = sqrt [ p(1-p) (1/n1 + 1/n2 )] = sqrt [0.2 * 0.8* 2/1000]

= 0.01789

Z = (p1991 -  p1986 )/ se0 = (0.26 - 0.14)/ 0.01789 = 6.707

so for alpha = 0.05 and upper directional Z - test

Zcritical = 1.645

so here Z > Zcritical , so we can reject the null hypothesis and conclude that market share of ibuprofen users have increased in year 1986 and 1991.

(c) yes, the test are related and as first test tell us that aspirin users are not same as earlier the letter test tells that ibuprofen users have increased. so Both test shows same kind of results.

Pain Reliever 1986 1991 Aspirin 45% 34% Acetaminophen 41% 41% Ibuprofen 14% 26%

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