Exercise Set Unit V

Testing Hypothesis About One Population Mean

1. This is a test of a single population mean when the population standard deviation is not known (large sample).  The P value approach

According to The "Long Life Battery Company" The average life of their automobile battery is 4.5 years. A researcher believes that this claim is not true and decides to construct a statistical test of the claim. He randomly selects 40 batteries from the company inventory and finds that the mean life of the 40 batteries is 3.98 years with sample standard deviation of 2.87 years. The researcher plans to test the null hypothesis at the 0.01 alpha level. 

(a) What is the null hypothesis? Answer:  population mean mu = 4.5 years/
(b) What is the alternative hypothesis? Answer: population mean mu not equal to 4.5 years
(c) What is the value of the standard error? Answer: 0.454
(d) What is the value of the test statistic? Answer: z =-1.15
(e) What is the p value for the test statistics? Answer: p =0.2518
(f) Is the p value smaller or larger than the alpha? Answer: larger
(g) What is the conclusion? Answer: Fail to reject the null hypothesis.

2. This is a test of a single population mean when the population standard deviation is not known (large sample). The classical approach

 It has been claimed that the mean weight of female students at Bainbridge College is 54.4 kg. Dr. Byrd does not believe the claim and sets out to show that the mean weight is not 54.4kg. To test the claim he randomly selects 100 female students from among the female student population at Bainbridge College. The results are; sample mean of 53.75 kg and sample standard deviation of 5.4 kg. Is this sufficient evidence  for Dr. Byrd to reject the claim at the 0.05 alpha level?

(a) What is the null hypothesis? Answer: Mean weight of female students at BC is 54.4 kg
(b) What is the alternative hypothesis? Answer: Mean weight of BC female students is not 54.4 kg
(c) What is the value of standard error? Answer: 0.54
(d) What is the value of the test statistic? Answer: z =-1.20
(e) What are the boundaries of the critical region? Answer: [-1.96, 1.96]
(f) Does the calculated statistic lie in the critical (reject) region? Answer: No
(g) Is this sufficient evidence for Dr. Byrd to reject the null hypothesis? Answer: No

3. This is a test of a single population mean when the population standard deviation is not known (small sample).  The "p" value or probability approach.

The Environmental Protection Agency (EPA) wants to show that carbon monoxide levels in a certain city are not in compliance with EPA standards. Specifically, the EPA wants to show that the mean level of carbon monoxide in the downtown portion of the city are higher that 4.9 parts per million (ppm) .Does a random sample of 22 readings ( sample mean = 5.1 ppm  and s=1.17 ppm ) present sufficient evidence to support the EPA's claim at the 0.05 alpha level?

(a) What is the null hypothesis? Answer: Mean PPM of carbon monoxide = 4.9
(b) What is the alternative hypothesis? Answer: mean ppm of carbon monoxide >4.9
(c) What is the value of the standard error? Answer: 0.2494
(d) What is the value of the test statistic? Answer: t=0.80
(e) Wow many df? answer: df=21
(f) What is the p value? Answer: p=0.216
(g) Is the "p" value smaller than alpha? Answer: No
(h) Is there sufficient evidence to support the EPA's claim that the carbon monoxide level is greater that 4.9ppm? Answer=NO 

 

Testing Hypothesis About Two Population Means (Dependent Samples)
(Matched Pairs)

Ten subjects with borderline high cholesterol levels were selected for a study. The study involved undergoing nutritional counseling. Cholesterol readings were taken before counseling and 4 months after counseling. The results are shown below.

subject                1     2       3       4      5       6      7     8      9      10
pre-counseling   295  279   250  235  255  290  310  260  275   240
counseling         265  266   245  240  230  230  235  250   250  215

  alpha =0.05

Note: The parameter of interest is: The mean of the paired differences in the cholesterol levels of the two groups. 

(a) Suppose the null hypothesis is:  Ub-Ua = 0
(b) What is the alternative hypothesis? Answer: Ub-Ua  > 0  
(c) What is the value of the difference in the means of the samples? Answer  26.3
(d) What is the value of the test statistic? Answer: t = 3.3946
(e) Is the test statistic in the critical region? Answer Yes
(f) What is the conclusion? Answer: Reject the null hypothesis
(g) What is the "p" value? Answer: p= 3.970146E-03
(h) Is P smaller than alpha? Answer: Yes
(f) What is your conclusion? Answer: reject the null hypothesis

Testing Hypothesis About Two Population Means (Independent Samples)

(Large samples, population standards not known, use z test)

1.It is generally believed that at Bainbridge College female students(U1) spend as much or more time (hrs) studying  that BC male students(U2). A study was conducted by a BC statistic class who collectively believed that just the opposite was true, and decide to test their claim at the 0.03 alpha level. The data collected for the study are shown below.

Female Students: 13.9, 17.3, 10.0, 21.2, 12.8, 13.7, 15.0, 17.3, 12.1, 13.5
                           17.9, 19.1, 18.2, 14.8, 19.1,   6.2,  10.6, 10.7, 23.1 15.7
                           14.1, 10.7, 17.2, 18.6, 16.3,  17.8, 16.2, 15.8,  7.5,  3.7

Male Students:    17.5, 27.9, 14.1,  9.9,  20.9,  19.6, 15.8, 21.8, 23.0, 20.6
                           20.5, 31.9, 24.0, 14.4, 22.5,  24.6, 12.0, 18.6, 22.8, 27.6
                           20.1, 18.5, 25.5, 20.0, 27.8, 13.2,  18.4, 21.6, 23.0, 24.5


(a) What is the null hypothesis? Answer: U1>=U2
(b) What is the alternative hypothesis? U1<U2
(c) What is the value of the test statistic? Answer: z = -4.22
(d) What is the "p" value? Answer: p < 0.0001
(e) Is P smaller than alpha? Answer: Yes
(f) What are the bounds for the critical region? Answer:  (-infinity , -1.88]
(g) does z lie in the critical (reject) region? Answer: Yes
(f) What should be the conclusion? Answer: Reject the null hypothesis

Testing Hypothesis About Two Population Means (Independent Samples)

(Small samples, population standards not known, populations normal Assume that population variances are equal use "t" test, pooled variance calculations.

1.It is generally accepted that  BC female students(U1) spend as much or more time  (hrs) studying  that BC male students(U2). A study was conducted by a BC statistic class who did not believe the generally accepted The class decided to test their claim  the 0.03 alpha level. The data collected for the study are shown below. Assume that the variances are equal.

Female Students: 13.9, 17.3, 10.0, 21.2, 12.8, 13.7, 15.0, 17.3, 12.1, 13.5
                           

Male Students:    17.5, 27.9, 14.1,  9.9,  20.9,  19.6, 15.8, 21.8, 23.0, 20.6
                           
a) What is the null hypothesis? Answer: U1>=U2
(b) What is the alternative hypothesis? U1<U2
(c) What is the value of the test statistic? Answer: t = -2.348
(d) What are the bounds for the critical region? Answer:  (-infinity , -2.007]
(e) Does the test statistic lie in the critical region? Answer: Yes
(f) What should be the conclusion? Answer: Reject the null hypothesis
(g) What is the "p" value? Answer: p = 0.0152
(h) Is P smaller than alpha? Answer: yes
(i) What should be the conclusion? Answer: Reject the null hypothesis