Question

It is widely accepted that people are a little taller in the morning than at night. Here we perform a test on how big the difference is. In a sample of 30 adults, the morning height and evening height are given in millimeters (mm) in the table below. Use this data to test the claim that, on average, people are more than 10 mm taller in the morning than at night. Test this claim at the 0.01significance level.

(a) The claim is that the mean difference (x - y) is more than 10 mm (μd > 10). What type of test is this? This is a two-tailed test.This is a left-tailed test.     This is a right-tailed test. (b) What is the test statistic? Round your answer to 2 decimal places. t d =   (c) Use software to get the P-value of the test statistic. Round to 4 decimal places. P-value =   (d) What is the conclusion regarding the null hypothesis? reject H0fail to reject H0      (e) Choose the appropriate concluding statement. The data supports the claim that, on average, people are more than 10 mm taller in the morning than at night.There is not enough data to support the claim that, on average, people are more than 10 mm taller in the morning than at night.     We reject the claim that, on average, people are more than 10 mm taller in the morning than at night.We have proven that, on average, people are more than 10 mm taller in the morning than at night.

AM Height (x) PM Height (y) (x - y) 1527 1518 9 1420 1408 12 1439 1432 7 1416 1407 9 1582 1575 7 1457 1444 13 1778 1765 13 1667 1655 12 1676 1664 12 1631 1620 11 1443 1429 14 1403 1392 11 1587 1573 14 1481 1465 16 1752 1742 10 1798 1780 18 1607 1592 15 1604 1585 19 1428 1419 9 1779 1770 9 1581 1572 9 1725 1715 10 1583 1574 9 1703 1696 7 1692 1685 7 1562 1553 9 1643 1631 12 1564 1556 8 1417 1407 10 1592 1579 13

Answer 1

a)  This is a right-tailed test.

b)

SAMPLE 1	SAMPLE 2	difference , Di =sample1-sample2	(Di - Dbar)²
1527	1518	9.000	4.551
1420	1408	12.000	0.751
1439	1432	7.000	17.084
1416	1407	9.000	4.551
1582	1575	7.000	17.084
1457	1444	13.000	3.484
1778	1765	13.000	3.484
1667	1655	12.000	0.751
1676	1664	12.000	0.751
1631	1620	11.000	0.018
1443	1429	14.000	8.218
1403	1392	11.000	0.018
1587	1573	14.000	8.218
1481	1465	16.000	23.684
1752	1742	10.000	1.284
1798	1780	18.000	47.151
1607	1592	15.000	14.951
1604	1585	19.000	61.884
1428	1419	9.000	4.551
1779	1770	9.000	4.551
1581	1572	9.000	4.551
1725	1715	10.000	1.284
1583	1574	9.000	4.551
1703	1696	7.000	17.084
1692	1685	7.000	17.084
1562	1553	9	4.55
1643	1631	12	0.75
1564	1556	8	9.82
1417	1407	10	1.28
1592	1579	13	3.48

	sample 1	sample 2	Di	(Di - Dbar)²
sum =	47537	47203	334.000	291.467

mean of difference ,    D̅ =ΣDi / n =   11.133                  
                          
std dev of difference , Sd =    √ [ (Di-Dbar)²/(n-1) =    3.1703                  
                          
std error , SE = Sd / √n =    3.1703   / √   30   =   0.5788      
                          
t-statistic = (D̅ - µd)/SE = (   11.13333333   -   10   ) /    0.5788   =   1.96

c)

Degree of freedom, DF=   n - 1 =    29  
  
p-value =        0.0300    [excel function: =t.dist.rt(t-stat,df) ]

d)

p-value>α , fail to reject null hypothesis

e)There is not enough data to support the claim that, on average, people are more than 10 mm taller in the morning than at night.