Question

In: Statistics and Probability

Pea aphids, Acyrthosiphon pisum, are small, wingless, sapsucking insects that live on plants. They evade predators...

Pea aphids, Acyrthosiphon pisum, are small, wingless, sapsucking insects that live on plants. They evade predators (such as ladybugs) by dropping off. A study examined the mechanism of aphid drops. Researchers placed aphids on a leaf positioned at four different heights (in centimeters, cm) above a surface covered with petroleum jelly. When a ladybug was introduced, the aphids dropped, and the petroleum jelly helped capture their landing posture (upright or not). Each aphid performed this experiment only once. The findings are given in the table.

Landing Posture	3 cm	5 cm	10 cm	20 cm
	Dropping Height
Upright	20	23	27	29
Not upright	10	7	3	1
Sample size	30	30	30	30

The null hypothesis "no relationship" says that in the population of aphids, the proportions that land upright are the same when the dropping height is 33 , 55 , 1010 , and 2020 cm.

The two‑way table contains expected cell counts if this hypothesis is true.

Landing Posture	33 cm	55 cm	1010 cm	2020 cm	Total
Upright	24.75	24.75	24.75	24.75	99
Not upright	5.25	5.25	5.25	5.25	5.25
Total	30	30	30	30	30

Question 1: Use the tables to calculate the value of the chi‑square statistic. (Enter your answer rounded to three decimal places.)

X^2 =

Question 2: Use software to calculate the related ?P‑value. (Enter your answer rounded to four decimal places.)

P =

Question 3: Which cells contribute the most to ?2χ2 ? Select the correct answer.

a) The largest contributions are that of aphids dropped from a height of either 33 cm or 2020 cm that do not land upright. They correspond to the two extreme dropping heights. The results suggest there is a relationship between dropping height and landing posture.

b) We cannot make a definitive conclusion about dropping height and landing posture from the data given. Further studies are necessary.

c) The largest contributions are that of aphids dropped from a height of either 55 cm or 1010 cm that do not land upright. They correspond to the two middle dropping heights. The results suggest there is a relationship between dropping height and landing posture.

d) The largest contributions are that of aphids dropped from a height of either 33 cm or 2020 cm that land upright. They correspond to the two extreme dropping heights. The results suggest there is a relationship between dropping height and landing posture.

Question 4: Is the ?P‑value significant at the 0.050.05 level? Choose the correct explanation.

a) The ?P‑value is not significant at the 0.050.05 level. Rejecting ?0H0 means that we have evidence that the proportion of upright landing is the same for all four dropping heights.

b) The ?P‑value is significant at the 0.050.05 level. Rejecting ?0H0 means that we have evidence that the proportion of upright landing is not the same for all four dropping heights.

c) The ?P‑value is not significant at the 0.050.05 level. This indicates that ?0H0 is correct and we have evidence that the proportion of upright landing is the same for all four dropping heights.

d) The ?P‑value is significant at the 0.050.05 level. This indicates that ?0H0 is correct and we have evidence that the proportion of upright landing is the same for all four dropping heights.

Expert Solution

1. X^2 = 11.255

2. P = 0.0104

3. a) The largest contributions are that of aphids dropped from a height of either 3 cm or 20 cm that do not land upright. They correspond to the two extreme dropping heights. The results suggest there is a relationship between dropping height and landing posture.

4. b) The ?P‑value is significant at the 0.050.05 level. Rejecting ?0H0 means that we have evidence that the proportion of upright landing is not the same for all four dropping heights.

		Col 1	Col 2	Col 3	Col 4	Total
Row 1	Observed	20	23	27	29	99
	Expected	24.75	24.75	24.75	24.75	99.00
	O - E	-4.75	-1.75	2.25	4.25	0.00
	(O - E)² / E	0.91	0.12	0.20	0.73	1.97
Row 2	Observed	10	7	3	1	21
	Expected	5.25	5.25	5.25	5.25	21.00
	O - E	4.75	1.75	-2.25	-4.25	0.00
	(O - E)² / E	4.30	0.58	0.96	3.44	9.29
Total	Observed	30	30	30	30	120
	Expected	30.00	30.00	30.00	30.00	120.00
	O - E	0.00	0.00	0.00	0.00	0.00
	(O - E)² / E	5.21	0.71	1.17	4.17	11.26

		11.26	chi-square
		3	df
		.0104	p-value

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orchestra answered 1 year ago

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