Question

Each box of Healthy Crunch breakfast cereal contains a coupon entitling you to a free package of garden seeds. At the Healthy Crunch home office, they use the weight of incoming mail to determine how many of their employees are to be assigned to collecting coupons and mailing out seed packages on a given day. (Healthy Crunch has a policy of answering all its mail on the day it is received.) Let x = weight of incoming mail and y = number of employees required to process the mail in one working day. A random sample of 8 days gave the following data.

x (lb)	14	22	15	6	12	18	23	25
y (Number of employees)	7	10	9	5	8	14	13	16

In this setting we have Σx = 135, Σy = 82, Σx² = 2563, Σy² = 940, and Σxy = 1530.

(e) If Healthy Crunch receives 14 pounds of mail, how many employees should be assigned mail duty that day? (Round your answer to two decimal places.)
8.7728 employees

(f) Find S_e. (Round your answer to three decimal places.)
S_e =

(g) Find a 95% for the number of employees required to process mail for 14 pounds of mail. (Round your answer to two decimal places.)

lower limit	employees
upper limit	employees

(h) Test the claim that the slope β of the population least-squares line is positive at the 1% level of significance. (Round your test statistic to three decimal places.)

t =

Find or estimate the P-value of the test statistic.

P-value > 0.250 0.125 < P-value < 0.250     0.100 < P-value < 0.125 0.075 < P-value < 0.100 0.050 < P-value < 0.075 0.025 < P-value < 0.050 0.010 < P-value < 0.025 0.005 < P-value < 0.010 0.0005 < P-value < 0.005 P-value < 0.0005

Conclusion

Reject the null hypothesis, there is sufficient evidence that β > 0. Reject the null hypothesis, there is insufficient evidence that β > 0.     Fail to reject the null hypothesis, there is sufficient evidence that β > 0. Fail to reject the null hypothesis, there is insufficient evidence that β > 0.

(i) Find an 80% confidence interval for β and interpret its meaning. (Round your answers to three decimal places.)

lower limit
upper limit

Interpretation

For each less pound of mail, the number of employees needed increases by an amount that falls within the confidence interval. For each additional pound of mail, the number of employees needed increases by an amount that falls outside the confidence interval.     For each additional pound of mail, the number of employees needed increases by an amount that falls within the confidence interval. For each less pound of mail, the number of employees needed increases by an amount that falls outside the confidence interval.

Answer 1

(i) lower limit = 0.341, upper limit = 0.685
Interpretation -> For each additional pound of mail, the number of employees needed increases by an amount that falls within the confidence interval.

(h) t = 4.295,
0.005 < P-value < 0.010
Reject the null hypothesis, there is sufficient evidence that β > 0.

(e) For x = 14, estimated number of employees = 8.77

(f) S_e = 2.017

(g) (It has not been mentioned here whether I have to find prediction interval or confidence interval, so I am calculating both.)
For prediction interval -> lower limit = 3.47, upper limit = 14.08
For confidence interval -> lower limit = 6.84, upper limit = 10.71