Question

Refer to the data set in the accompanying table. Assume that the paired sample data is a simple random sample and the differences have a distribution that is approximately normal. Use a significance level of 0.01 to test for a difference between the weights of discarded paper? (in pounds) and weights of discarded plastic? (in pounds).

Household	Paper	Plastic
1	9.41	3.36
2	15.09	9.11
3	6.83	3.57
4	7.98	6.09
5	17.65	11.26
6	7.57	5.92
7	13.31	19.7
8	9.19	3.74
9	12.32	11.17
10	13.61	8.95
11	16.08	14.36
12	6.44	8.4
13	11.08	12.47
14	9.55	9.2
15	2.8	5.92
16	11.42	12.81
17	20.12	18.35
18	9.45	3.02
19	6.38	8.82
20	6.98	2.65
21	13.05	12.31
22	16.39	9.7
23	6.96	7.6
24	6.67	6.09
25	5.86	3.91
26	7.72	3.86
27	8.72	9.2
28	11.36	10.25
29	6.16	5.88
30	6.33	3.86

t=?

p=?

Answer 1

x1=paper , x2 = plastic

First enter Data into EXCEL

We have to find the sample mean.

Excel command is =AVERAGE(Select data)

sample mean =$\bar{x1}$ = 10.08

sample mean =$\bar{x2}$ = 8.38

Now we have to find sample standard deviation.

Excel command is =STDEV(Select data)

standard deviation = s1 = 4.11

standard deviation = s2 = 4.40

n1 = 30

n2 = 30

$\bar{x1}$= 10.08

$\bar{x2}$ = 8.38

s1 = 4.11

s2 = 4.40

Null and alternative hypothesis is

H0 :u1 = u2

Vs

H1 :u1? u2

Level of significance = 0.01

Before doing this test we have to check population variances are equal or not.

Null and alternative hypothesis is

$H0:\sigma _{1}^{2}=\sigma _{2}^{2}$

Vs

$H1 : :\sigma _{1}^{2 } ? \sigma _{2}^{2}$ ? $\sigma _{2}^{2}$

Test statistic is

F = Larger variance / Smaller variance = 19.36 / 16.8921 = 1.146

Degrees of freedoms

Degrees of freedom for numerator = n1 - 1 = 30 - 1 = 29

Degrees of freedom for denominator = n2 - 1 = 30 - 1 = 29

Critical value = 2.42            ( using f-table )

F test statistic < critical value    we fail to reject null hypothesis.

Conclusion: Population variances are equal.

So we have to use pooled variance.

Formula

$t=\frac{\bar{x1}-\bar{x2}}{Sp*\sqrt{\frac{1}{n1}+\frac{1}{n2}}}$

$Sp = \sqrt{\frac{(n1-1)s1^{2}+(n2-1)s2^{2}}{n1+n2-2}}$

30-1)16.8921+(30-1)19.36 30+30-2

$t=\frac{10.08 - 8.38 }{ 4.257 *\sqrt{\frac{1}{30}+\frac{1}{30}}}$

$t=1.546$

? = 0.01 , d.f = n1 + n2 – 2 = 3 0+ 30 - 2 = 58

p-value = 0.127 ( using t table )

p-value > ? ,Failed to Reject H0