Hypothesis testing for two populations: statistical analysis môn Xác suất thống kê| Trường Đại học Ngoại Thương

1. Shirley Brown, an agricultural economist, wants to compare cow manure and turkey dung
as fertilizers. Historical y, farmers had used cow manure on their cornfields. Recently, a major
turkey farmer offered to sel composted turkey dung at a favorable price. The farmers decided
that they would use this new fertilizer only if there was strong evidence that productivity
increased over the productivity that occurred with cow manure. Shirley was asked to conduct
the research and statistical analysis in order to develop a recommendation to the farmers. Tes
at the significance level of 0.05; n1 = 25  , 𝑥 = 1100, 𝜎1  = 115, 𝜎2
2=400 (𝑐𝑜𝑤), n2 = 25, 𝑥2 2= 625 (turkey)
2. You have been asked to determine if two different production processes have different mean
numbers of units produced per hour. Pocess 1 has a mean defined as 𝜇1 and process 2 has a
defined as 𝜇2. The null and alternative hypotheses are as fol ows: H0: 𝜇1− 𝜇2= 0; H1 𝜇1− 𝜇2> 0
Using a random sample of 25 observations from process 1 and 28 observations from process
2, and the known variance for process 1 equal to 900 and the known variance for process 2
equal to 1,600. Can you reject the nul hypothesis using a probability of Type I error =0.05 if 𝑥1 = 50, 𝑥  2 = 60
a) The process means are 50 and 60?
b) The difference in process mean is 20?
c) The process means are 45 and 50?
d) The difference in process is 15?
3. You have been asked to determine if two different production processes have different
mean numbers of units produced per hour. Pocess 1 has a mean defined as 𝜇1 and process 2
has a mean defined as 𝜇2. The nul and alternative hypotheses are as fol ows: H0: 𝜇1−
𝜇2≤ 0; H1: 𝜇1− 𝜇2> 0
The process variances are unknown but assumed to be equal. Using a random sample of 25
observations from process 1 and 36 observations from process 2, the sample means are 56
and 50 for populations 1 and 2 . Can you reject the nul hypothesis using a probability of Type I error =0.05 if
a) The sample standard deviation from process 1 is 30 and from process 2 is 28?
b) The sample standard deviation from process 1 is 22 and from process 2 is 33?
c) The sample standard deviation from process 1 is 30 and from process 2 is 42?
d) The sample standard deviation from process 1 is 15 and from process 2 is 36?
4. Test the hypotheses H0: 𝑝1− 𝑝2= 0; H1: 𝑝1− 𝑝2< 0
Using the fol owing statistics from random samples: a) 𝑝1 = 0.42, n1 = 500; 𝑝  2 = 0.5, n2 = 600 b) 𝑝1 = 0.6, n1 = 500; 𝑝  2 = 0.64, n2 = 600 c) 𝑝1 = 0.42, n1 = 500; 𝑝  2 = 0.49, n2 = 600 d) 𝑝1 = 0.25, n1 = 500; 𝑝  2 = 0.34, n2 = 600 e) 𝑝1 = 0.39, n1 = 500; 𝑝  2 = 0.42, n2 = 600
5. Random samples of 900 people in the United States and in Great Britain indicated that 60%
of the people in the United States were positive about the future economy, while 66% of the
people in Great Britain were positive about the future economy. Does this provide strong
evidence that the people in Great Britain are more optimistic about the economy? (Significance level of 0.05) 6. . Test the hypotheses H0 2 : =𝜎1
𝜎2 2; H1: 𝜎21> 𝜎2 2
Using the fol owing statistics from random samples:
a) 𝑠12= 125, n1 = 45; 𝑠22= 51, n2 = 41
b) 𝑠12= 125, n1 = 45; 𝑠22= 235, n2 = 44
c) 𝑠12= 134, n1 = 48; 𝑠22= 51, n2 = 41
d) 𝑠12= 88, n1 = 39; 𝑠22= 167, n2 = 25
7. A university research team was studying the relationship between idea generation by groups
with and without a moderator. For a random sample of four groups with a moderator the mean
number of ideas generated per group was 78.0, and the standard deviation was 24.4. For a
random sample of four groups without a moderator the mean number of ideas generated per
group was 63.5, and the standard deviation was 20.2. Test the assumption that the two
population variances were equal against the alternative that the population variance is higher for groups with a moderator.