3.) You go out and collect the following estimates of earthworms / acre: 54,276

Question

3.) You go out and collect the following estimates of earthworms / acre: 54,276 57,378 51,108 66,190 66,232 59,018 57,159 These data yield the following: ¯y = 58,766, and s = 5,689.606

a) Construct a 90% CI for these data.

b) Construct a 95% CI for these data.

c) Darwin once estimated that an acre of soil had about 50,000 worms in it. Is his estimate consistent with the data above? (Historical note: His estimate was considered way too high in his day).

4) Consider the results of 3(b). Notice that all the data fit within the 99% confidence interval. Is this usually the case (in other words, will a 99% CI contain most of the observations)?

Hint & caution: a lot of people get this wrong. Here's a hint: suppose you had measured the worms in 6000 acres (instead of just 7). What happens to the confidence interval? If you're not sure, substitute 6,000 for 7 in your calculation for (b) to see what happens.

I really mostly need help with part 3c and question 4 (last two questions)

Explanation / Answer

Answer:

3.) You go out and collect the following estimates of earthworms / acre: 54,276 57,378 51,108 66,190 66,232 59,018 57,159 These data yield the following: ¯y = 58,766, and s = 5,689.606

a) Construct a 90% CI for these data.

Confidence Interval Estimate for the Mean

Data

Sample Standard Deviation

5689.606

Sample Mean

58766

Sample Size

7

Confidence Level

90%

Intermediate Calculations

Standard Error of the Mean

2150.468933

Degrees of Freedom

6

t Value

1.9432

Interval Half Width

4178.7488

Confidence Interval

Interval Lower Limit

54587.25

Interval Upper Limit

62944.75

b) Construct a 95% CI for these data.

Confidence Interval Estimate for the Mean

Data

Sample Standard Deviation

5689.606

Sample Mean

58766

Sample Size

7

Confidence Level

95%

Intermediate Calculations

Standard Error of the Mean

2150.468933

Degrees of Freedom

6

t Value

2.4469

Interval Half Width

5262.0079

Confidence Interval

Interval Lower Limit

53503.99

Interval Upper Limit

64028.01

c) Darwin once estimated that an acre of soil had about 50,000 worms in it. Is his estimate consistent with the data above? (Historical note: His estimate was considered way too high in his day).

Both 90% CI and 95% CI does not contain 50,000. Both lower limits are higher than 50000. Therefore his estimate worms does not consistent with the Darwin estimated

4) Consider the results of 3(b). Notice that all the data fit within the 99% confidence interval. Is this usually the case (in other words, will a 99% CI contain most of the observations)?

99% confidence interval means we are 99% confident that the true population mean will contain in it. It is not usually 99% CI contain most of the observations.

99% confidence interval is wider than the 95% confidence interval. Increasing the confidence level of a confidence interval will only result in larger intervals. That's not about accuracy or precision but rather about how much confidence that we are to take about the true values are in.But when sample size increases , the width of the interval decreases and precision increases.

Confidence Interval Estimate for the Mean

Data

Sample Standard Deviation

5689.606

Sample Mean

58766

Sample Size

7

Confidence Level

90%

Intermediate Calculations

Standard Error of the Mean

2150.468933

Degrees of Freedom

6

t Value

1.9432

Interval Half Width

4178.7488

Confidence Interval

Interval Lower Limit

54587.25

Interval Upper Limit

62944.75

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