Question

You are required to write a concise report with all your answers in doc or pdf file. You should include tables or figures from Excel in your report if needed. All data sets were simulated using a statistical software. You can also find Excel temples in the attachment with my announcement, which helps you a lot!

Part A. A researcher intended to investigate the potential difference in BMI between men and women. He also tried to evaluate the effectiveness of a new physical exercise program in reducing BMI. Table 1 summarized the BMI data. At the beginning of the study (Day 0), 64 men and 49 women were included in the study and their BMI values were measured (defined as BMI_M0 and BMI_F0, respectively). After joining the new physical exercise program for 180 days, the participants's BMI values were measured again (defined as BMI_M180 and BMI_F180, respectively). Only BMI_M180 data were included in Table 1. BMI_M0 and BMI_M180 were paired by row, i.e BMI_M0 and BMI_M180 values at the same row were from the same male participant. In addition, Table 1 included BMI_F0X, a data set of women BMI without any intervention from another researcher.

To answer the following questions, you need to decide which t test is needed, i.e. you need to decide (1) whether the t test is two-sided or one-sided; (2) whether the t test is paired or independent with Equal Variances or independent with Unequal Variances. For checking Equal Variances assumption between two independent samples, please refer to the lecture or Excel Assignment 2.

BMI_M0	BMI_F0	BMI_M180	BMI_F0X
29.6	27.1	22.9	21.8
19.9	29.8	25.6	28.5
21.5	32.8	23.8	21.8
15.8	23.5	27.7	26.6
22.9	27.8	24.2	23.9
28.1	20.1	24	27.4
27.5	26.5	22.3	26.5
25.3	18	22.9	30.7
33.4	12.9	27.5	28.4
28.5	24.9	17.6	24.9
30.4	31.5	22.5	25.3
30.9	16.2	22.4	17.8
36.4	29.9	27.6	25.3
22.5	20.1	20.7	23
25.4	18.7	20.7	25.1
34	25.6	23.8	28.5
31.9	23.8	22.5	24.6
28.4	19.4	30.5	28.2
30	33.4	18.5	27
32.4	22.4	20.6	23.7
34.7	30.4	25.2	25.2
23.7	17.4	20	22.2
21	25.8	19.1	27.5
35.2	27.2	27	22.4
22.9	28.3	24.7	23.3
24.3	21.8	24.9	23.5
30.1	25.6	27.4	20.1
23.5	27.4	26.2	26.5
26.1	18.4	24.1	24.1
21.3	20.6	30.6	23.4
33.6	27.8	23.9	22.6
19.4	27.3	24.1	24.8
29.5	19.2	26	31.3
29.2	17.8	24.3	28.8
27.9	33.5	17.3	27.3
24.3	23.8	23.4	24.3
25.7	18.6	26.4	27
28.1	22.8	18.9	23.6
31	14.7	23.7	28.3
25.5	17.7	26	22.3
34.1	29.5	25.2	23.1
30.6	23.6	25.7	24.6
25.6	24	30.1	31.1
30.4	27.7	18.4	23.7
25.22	25.9	26.1	22.8
34.7	32	29.6	28.1
29.2	25.4	28.6	26.7
21.6	23.8	23.6	26.2
24.5	27.3	17.8	28.4
28		24.7
19.9		24.2
29.8		23.1
27.6		23
34.2		21.7
22.2		26.6
27.1		20.6
28		22.4
20.2		22.6
22.8		26.6
30.9		26
25.6		29.4
26.8		24.7
26		20.4
31.5		26.8

Q1. Was there any statistically significant difference between BMI_M0 and BMI_F0? The significance level was 0.05. Please indicate which t test is used, what the p value is from your test and what your decision is.

Q2. Was there any statistically significant difference between BMI_M0 and BMI_F0X?

The significance level was 0.05. Please indicate which t test is used, what the p value is from your test and what your decision is.

Q3. Did the new physical exercise program significantly reduce BMI in men after 180-days exercise (i.e. comparing BMI_M0 and BMI_M180)? The significance level was 0.05. Please indicate which t test is used, what the p value is from your test and what your decision is.

Part B. In regard to a clinical index (categorical), the expected frequencies for LOW, MEDIUM, and HIGH were 20%, 30%, and 50%, respectively. A researcher evaluated 200 patients, and found that 83, 44, and 73 patients were LOW, MEDIUM, and HIGH, respectively. Were the observed frequencies significantly different from the expected frequencies? The significance level was 0.05. Please indicate which test is used, what the p value is from your test and what your decision is. (6 points)

Part C. A researcher intended to investigate the potential association between Age Group and Smoking Status. He collected data from 575 participants. The data was summarized in Table 2. Was the data in support of a statistically significant association between Age Group and Smoking Status? The significance level was 0.05. Please indicate which test is used, what the p value is from your test and what your decision is. (6 points)

Table . Age Group and Smoking Status

	Non smoker	Occasional smoker	Frequent smoker
Younger than 35	23	45	35
35~50-years-old	33	44	65
51~65-years-old	43	63	90
Older than 65	72	27	35

Answer 1

Question 1 solved completely

Q1. Was there any statistically significant difference between BMI_M0 and BMI_F0? The significance level was 0.05. Please indicate which t test is used, what the p value is from your test and what your decision is.

We first find the mean and std. deviation of each sample. as shown below using excel.

E F 2 G н BMI_MO BMI_FO BMI_M180 BMI_FOX Mean =AVERAGE(A2:A65) =AVERAGE(B2:B65) =AVERAGE(C2:C65) =AVERAGE(D2:D65) Std. Dev =STDEV.S(A2:A65) =STDEV.S(B2:B65) ESTDEV.S(C2:C65) ESTDEV.S(D2:D65) No. of observation =COUNT(A2:A65) =COUNT(B2:B65) =COUNT(C2:C65) =COUNT(D2:D65) Mean Std. Dev No. of observation 27.3175 4.6009 24.2796 5.123 24.0531 3.2309 A B C D 1 BMI_MOBMI_FO BMI_M181 BMI_FOX 29.6 27.1 22.9 21.8 19.9 29.8 25.6 28.5 21.5 32.8 23.8 21.8 15.8 23.5 27.7 26.6 22.9 27.8 24.2 23.9 28.1 20.1 24 27.4 27.5 26.5 22.3 26.5 25.3 18 22.9 30.7 33.4 12.9 27.5 28.4 28.5 24.9 17.6 24.9 12 30.4 31.5 22.5 25.3 30.9 16.2 22.4 17.8 36.4 29.9 27.6 25.3 22.5 20.1 20.7 16 25.4 18.7 20.7 25.1 25.351 2.8337 64 49 64 49 11 13 14 15 23 A ac

We use a
two sided test
T test that is independent with equal variance.

Part 1 : Test if the variance are equal.

The objective in this question is to test whether to the two population variances are equal or not. Given sz=21.1682808 ni = 64 sz= 26.245129 n2 = 49 Step 1 : Hypothesis Ho : = 1 H:+1 Step 2: Decision Rule Finding the numerator and denominator degrees of freedom vi = n1-1= 64 -1=63 U2 = n2-1 = 49 -1 = 48 Finding the Fvalue Fla/2.01, v2] = F0.05,63,48) = 0.6429 Excel formula : FDIST(0.05,63,48) F(1 - c.01.02) = (a/2.02.01) -= 1.5802 F(0.05,48,63) Excel formula : 1/1/FDIST(0.05,48,63) Reject Ho, F> 1.5802 Reject H., F< 0.6429

Step 3: Test Statistics F__ $1 – 21.1682808 s 26.2451 29 0.8066 Step 4: Pvalue method Using test statistics and degree of freedom, we find the pvalue using the table pvalue = 0.4208 Step 5: Conclusion Critical value method : Since the test statistics( 0.8066) lies between 0.6429 and 1.5802, we fail reject the null hypothesis. Therefore, there is not enough evidence to claim that the population variances differ from each other 1 Pvalue method : Since the pvalue(0.4208) is greater than 0.1,we fail reject the null hypothesis

Part 2 : Hypothesis if there is a significant difference in the mean of the two samples.

The provided sample means are shown below: X1 = 27.3175 Xa = 24.2796 Also, the provided sample standard deviations are: Si = 4.6009 82 = 5.123 and the sample sizes are n1 = 64 and no = 49. (1) Null and Alternative Hypotheses The following null and alternative hypotheses need to be tested: Ho: Mi = 112 Ha: 1 # This corresponds to a two-tailed test, for which a t-test for two population means, with two independent samples, with unknown population standard deviations will be used. (2) Rejection Region Based on the information provided, the significance level is a = 0.05, and the degrees of freedom are df = 111. In fact, the degrees of freedom are computed as follows, assuming that the population variances are equal: Hence, it is found that the critical value for this two-tailed test is te = 1.982, for a = 0.05 and df = 111. The rejection region for this two-tailed test is R = {t : t > 1.982).

(3) Test Statistics Since it is assumed that the population variances are equal, the t-statistic is computed as follows: X1 - X -1.8+(12-1)3 ( + V ) 11, +Ty - 2 27.3175-24.2796 (64-1)4.60092+(49-1]5.123 / 1 1 - = 3.311 +1) 64+49 ( (4) Decision about the null hypothesis Since it is observed that t = 3.311 >te = 1.982, it is then concluded that the null hypothesis is rejected Using the P-value approach: The p-value is p = 0.0013, and since p = 0.0013 < 0.05, it is concluded that the null hypothesis is rejected. (5) Conclusion It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that population mean Mi is different than 42, at the 0.05 significance level.