Pineapple Farm The Pineapple Problem

By Tearies Silvani At the Drole Pineapple Company, managers are always interested in the sizes of the pineapples grown in the company's fields. The weights of pineapples grown in the Drole Fields were approximately normally distributed with a standard deviation of 2.5 ounces.

*Last year, the pineapples harvested had a mean weight of 31 ounces. Suppose a single pineapple is selected at random from last year's crop. What is the approximate probability that it weighs more than 35 ounces? Now suppose that two pineapples are selected at random from last year's crop. What is the approximate probability that it weighs more than 35 ounces ? In order to convince local distributors to sell your pineapples, you decide to show them a few of your bigger fruits. If you select pineapples at random from last year's crop, how many pineapples might you have to inspect before you find one that weighs more than 35 ounces? B. Perform your stimulation twice, clearly illustrating your procedure and the results. Begin on line 127.

First the selection of 50 pineapples:

- Numbers we looked at

- Numbers we've selected for sample 50

43909 99477 25330 64359 40085 16925 85117 36071 15689 14227 06565 14374 13352

49367 81982 87209 36759 58984 68288 22913 18638 54303 00795 08727 69051 64817

87174 09517 84534 06489 87201 97245 05007 16632 81194 14873 04197 85576 45195

96565 68732 55259 84292 08796 43165 93739 31685 97150 45740 41807 65561 33302

07051 93623 18132 09547 27816 78416 18329 21337

9 5 3 8 5 3 5 2 6 4 4 1 2 5 4 2 8 3 9 7 6 7 5 3 9 9 5 6 9 3 8 5 5 3 9 0 3 7 8 0 4 5 3 5 3 0 7 4 2 7

Now the selection for the best 12 pineapples:

- Numbers from the 50 selected pineapples

- Numbers we've selected to show the local distributors

95385 35264 41254 28397 67539 95693 85539 03780 45353 07427

8 2 1 2 7 3 6 5 0 0 5 4

Find the approximate probability that your first suitable pineapple (weighing 35 or more ounces) is obtained within the first 3 pineapples. Managers were eager to test a new irrigation system, and did so during this year's production cycle.

How large of a sample would you need to take to estimate the mean weight of the pineapples produced to within one ounce at a 95% confidence level? An SRS of 50 pineapples taken from this year's crop had a mean weight of 31.6 ounces.

Use a 95% confidence level to determine whether this sample provides evidence of a change in the mean weight of pineapples produced. Explain you reasoning. Did the new irrigation system cause an increase in the mean weights of pineapples produced? Explain your answer. THE

END *Normalcdf(l, u, mean, standard deviation)

L- lower bound

U- upper bound

2ndDISTR: Normalcdf(35, 1E99, 31, 2.5) = .0548

The approximate probability that a pineapple weighs more than 35 ounces from last year's crop is .0548. *The Binomial model counts the number of successes in n trials.

Binompdf(n, p, x)

n- number of trials

p- probability of success

x- number of success in trial(s)

2ndDISTR: Binompdf(2, .0548, 1) = .1036

The approximate probability of exactly one of the two pineapples that weighs more than 35 ounces is 10.36%. *The Geometric model counts the number of trials until the first success.

Genompdf(p, x)

p- probability of success

x- number of trials until the first success

2ndDISTR: Genompdf(.0548, 3) = .049

The approximate probability that the first suitable pineapple (weighing 35 or more ounces) is obtained within the first three pineapples is .049. *Find the Margin of Error: ME = z(SD/SQ(n))

z- the z-score

sd- standard deviation

n- sample number

ME(1) = 1.96(2.5/SQ(n))

1/1.96= 2.5/SQ(n)

2.5 x .51= 2.5/SQ(n) x 2.5

4.9^2 = SQ(n)^2

24.01 = n

At a 95% confidence interval level, we would need a sample size of 24 pineapples to estimate the mean weight of the pineapples produced to within one ounce. *Confidence Interval estimates a populations parameter.

Stat: Zinterval:

- SD- 2.5

- x- 31.6

- n- 50

- c-level- .95

- calculate

- (30.91, 32.29)

At a 95% confidence level, there is no change in the mean weight because the mean weight of the sample of 50 pineapples taken from this year's crop had a mean weight of 31.6 ounces which falls between the confidence interval (30.91, 32.29). The null hypotheses is Ho: µ = 31

The alternative hypothese is Ha: µ > 31

STAT: TESTS: Z-Test: Input Data

µ- 31

SD- 2.5

x- 31.6

n- 50

µ: µ > 31

Calculate

z: 1.697

p: .0448

At a 95% confidence level, we can reject the null hypotheses (Ho: µ = 31) because of the p-value (.0448) being less than the sigificance value of .05. This means that the mean weigt of pineapples is actually larger than 31 ounces. A. Design a stimulation using a random digits table to mimic the search for a suitable pineapple.

We decided to gather every sixth pineapple that has grown in last year's crop until we collect a total of 50 pineapples. Then we took every fourth pineapple until a total of 12 pineapples were collected and showed the local distributors

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