Question

A new cereal called Tops is being tested. In the data table that follows, x is the time on the shelf in days and y is the moisture content, measured on a scale of 1 to 10.

Time on shelf (days)   0    6       10      16      24      30      37

Moisture content       2.9 3.2     3.5     3.6    3.9     4.2     4.6

Is the sample correlation coefficient r significant at the 0.05 level? Assume a one-sided (right-tail) alternate hypothesis. Find the equation of the least-squares line. What is the predicted value of the moisture content at 20 days on the shelf? Find a 90% confidence interval for the moisture content in part c above. Find a 90% confidence interval for the expected value of the moisture content at 20 days. Can we conclude, at α = 0.01, that the slope is positive? Why? Find a 95% confidence interval for β0, the y-intercept.

Answer 1

From the given data

	Time (X)	Moisture (Y)	X^2	Y^2	XY
1	0	2.9	0	8.41	0
2	6	3.2	36	10.24	19.2
3	10	3.5	100	12.25	35
4	16	3.6	256	12.96	57.6
5	24	3.9	576	15.21	93.6
6	30	4.2	900	17.64	126
7	37	4.6	1369	21.16	170.2
Total:	123	25.9	3237	97.87	501.6

c) The  predicted value of the moisture content at 20 days on the shelf is

Y = 2.94044+ 0.04323(20) = 3.805

d) The 95% confidence interval of Y-hat is (3.7284, 3.8816)

e)  The 95% prediction interval of Y-hat is (3.5918, 4.0181)