Question

florence is the owner of pltz grove, an orchard with 1,800 cherry trees. a late frost this season has caused some of the trees to not produce fruit, and florence wants to estimate how this will affect the total crop. she asks each of her workers to randomly check 15 trees to see whether or not the tree is producing fruit. here are the data from one workers sample, with y indicating a tree that is producing fruit and n indicating a tree that is not producing fruit: n n y y y n n y n y n y y n y based on the results of the sample, how many of the trees in the orchard should florence expect to not be producing fruit? trees

Explanation:

Step1: Count non-producing trees in sample

Count the number of 'n's in the sample: n, n, y, y, y, n, n, y, n, y, n, y, y, n, y. Let's count: first two n's, then two n's (positions 6,7), then n (position 9), n (position 11), n (position 14). So total n's: 2 + 2 + 1 + 1 + 1 = 7? Wait, let's list all:

1: n

2: n

3: y

4: y

5: y

6: n

7: n

8: y

9: n

10: y

11: n

12: y

13: y

14: n

15: y

Now count n's: positions 1,2,6,7,9,11,14. That's 7 n's. So in a sample of 15 trees, 7 are not producing.

Step2: Find proportion of non-producing

Proportion \( p = \frac{\text{number of non-producing in sample}}{\text{sample size}} = \frac{7}{15} \)? Wait, wait, wait, let's recount. Wait 1,2,6,7,9,11,14: that's 7? Wait 1 (n), 2 (n) – 2. Then 6 (n),7 (n) – 2 more (total 4). Then 9 (n) – 5. 11 (n) – 6. 14 (n) – 7. Yes, 7 n's. So sample size is 15, non-producing is 7.

Step3: Estimate total non-producing in orchard

Total trees in orchard: 1800. So expected non-producing is \( 1800 \times \frac{\text{number of non-producing in sample}}{\text{sample size}} = 1800 \times \frac{7}{15} \). Let's calculate: 1800 divided by 15 is 120. Then 120 times 7 is 840? Wait, wait, no: 1800/15 = 120. 120 * 7 = 840? Wait, but wait, maybe I miscounted the n's. Wait let's check again:

Sample data: n, n, y, y, y, n, n, y, n, y, n, y, y, n, y.

Let's list each:

1: n

2: n

3: y

4: y

5: y

6: n

7: n

8: y

9: n

10: y

11: n

12: y

13: y

14: n

15: y

Now count n's: 1,2,6,7,9,11,14. That's 7 n's. So 7 out of 15.

Then total expected: 1800 (7/15) = (1800/15)7 = 120*7 = 840? Wait, but wait, maybe I made a mistake in counting. Wait, let's count again:

n occurs at positions 1,2,6,7,9,11,14. That's 7 times. So 7 non-producing in 15. So proportion is 7/15. Then total non-producing is 1800 * (7/15). Let's compute:

1800 ÷ 15 = 120. 120 * 7 = 840. Wait, but wait, maybe I miscounted the n's. Wait, let's check the sample again:

Wait the sample is: n n y y y n n y n y n y y n y. Let's count the number of n's:

First two: n, n (2)

Then after three y's: n, n (positions 6,7: 2 more, total 4)

Then after y (position 8): n (position 9: 1 more, total 5)

Then after y (position 10): n (position 11: 1 more, total 6)

Then after two y's (positions 12,13): n (position 14: 1 more, total 7)

Yes, 7 n's. So 7 out of 15. Then 1800 (7/15) = 840. Wait, but let me check again. Wait 15 trees, 7 non-producing. So the proportion is 7/15. Multiply by 1800: 1800 7 /15 = (1800 /15)7 = 1207 = 840. So Florence should expect 840 trees to not be producing fruit.

Wait, but wait, maybe I made a mistake in counting the n's. Let's list all 15 elements:

1. n

1. n

1. y

1. y

1. y

1. n

1. n

1. y

1. n

1. y

1. n

1. y

1. y

1. n

1. y

Now count n's: 1,2,6,7,9,11,14. That's 7. So 7 non-producing in 15. So the calculation is correct.

Answer:

840