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 Σ*x* = 135, Σ*y* = 82,
Σ*x*^{2} = 2563, Σ*y*^{2} = 940, and
Σ*xy* = 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.)

(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

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

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