Question

Psychologist wants to determine if there is a linear relationship between the number of hours a person goes without sleep and the number of mistakes he/she makes on a simple test. The following data is recorded.

Hours without sleep	32	38	48	24	46	35	30	34	42
Number of Mistakes	6	8	13	5	7	6	5	8	12

Hours without sleep=x, Number of mistakes=y

1. Find the equation of regression line. Round slope and y-intercept to nearest thousandth.  

2. Find the residual when the number of hours is 32 hours.  Analyze the result.  

3. Apply critical value test (not hypothesis testing).  Show inequality. By using critical value test, is there linear relation exists?  

Answer 1

Here in this scenario Psychologist wants to determine if there is a linear relationship between the number of hours a person goes without sleep and the number of mistakes he/she makes on a simple test.

We have to use simple linear regression model to find out the correlation and linear regression equation as below,

The independent variable is XX, and the dependent variable is YY. In order to compute the regression coefficients, the following table needs to be used:

	X	Y	X*Y	X2	Y2
	32	6	192	1024	36
	38	8	304	1444	64
	48	13	624	2304	169
	24	5	120	576	25
	46	7	322	2116	49
	35	6	210	1225	36
	30	5	150	900	25
	34	8	272	1156	64
	42	12	504	1764	144
Sum =	329	70	2698	12509	612

Based on the above table, the following is calculated:

Based on the above table, the following is calculated: n 329 X X; 36.555555555556 < n 9 n 70 Y = ΣΥ; 7.7777777777778 n 9 O n n SSxx = x ΣΧ; 12509 – 3292/9 = 482.22222222222 п i=1 i=1 2 n n 1 SSYY = ΣΥ? Y; = 612 – 702 /9 = 67.555555555556 п i=1 i=1 < SSxy = x;Y 3xx -(*) () = 2698 – 329 x 70/9 = 139.111111 o Therefore, based on the above calculations, the regression coefficients (the slope m, and the y- intercept n) are obtained as follows: m = SSXY SSxx 139.11111111111 482.22222222222 0.2885 n = Y - X : m = 7.7777777777778 – 36.555555555556 x 0.2885 -2.7677 Therefore, we find that the regression equation is: ( Y = -2.7677 + 0.2885 X Graphically: O Scatter Plot and Regression Line 13.40 III 11.40 9.40 7.40 o 5.40 3.49.2021.2023.205.2027 2029 2031.2063.2035 2037.2039.2041,2043.2045.2047.2049.201.20 Regression equation: Y = -2.7677 + 0.2885 * X The correlation coefficient r is computed using the following expression: o r = SSXY VSSxx SSyy Therefore, based on this information, the sample correlation coefficient is computed as follows o r = SSXY V SSxx SSYY 139.111 482.222 x 67.556 = 0.771 which completes the calculation. = The following needs to be tested: H:p=0 ( HA:=0 where p corresponds to the population correlation. O The sample size is n = 9, so then the number of degrees of freedom is df = n - 2 = 9 – 2 = 7 The corresponding critical correlation value re for a significance level of a = 0.05, for a two- tailed test is: = rc = 0.666 Observe that in this case, the null hypothesis H. :p O is rejected if |r|>rc = 0.666. Based on the sample correlation provided, we have that |r| = 0.771 > rc = 0.666, from which is concluded that the null hypothesis is rejected.

Now ,

1. the equation of regression line. Round slope and y-intercept to is given below,

Yhat = -2.768 + +2.88 X  

Using the above Regression equation we can predict the number of mistek based on sleep.

2. The residual when the number of hours is 32 hours.

The residual means difference between the observed value and predicted value.

e = yi - Yhat

The predicted value of y when x is 30 is,

Yhat = -2.768 + 0.288 * 30

Yhat = 5.887.

The predicted value of number of mistek is 5.887 when sleep is 30 .

e = 5 - 5.887

e = -0.887.

The residual is -0.887.

3. Apply critical value test (not hypothesis testing).  Show inequality. By using critical value test, is there linear relation exists is given above.

Hope it helps .

Thank you.