You go out and collect the following estimates of earthworms / acre: 54,276 57,3

Question

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 60% CI for these data.

b) Construct a 99% 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.

5) The average height of women is y = 64 inches, the average height of men is y = 67 inches. For both the standard deviation is about s = 3 inches. (a) Suppose you take a sample of 6 women and 6 men. Construct a 95% CI for both. Do the confidence intervals overlap? (b) Now repeat using a sample of 123 men and 123 women. Do the confidence intervals overlap? (c) Can you explain what happened? Why is sample size important if you're trying to find differences between groups?

Explanation / Answer

Question 3 .

(a) 60% CI =   y +- tdF,0.40 (s/sqrt(n)

= 58766 +- 0.9057 * 5689.61/sqrt(7)

= (56818.31, 60713.69)

99% CI = y +- tdF,0.01 (s/sqrt(n)

= 58766 +- 3.7074 * 5689.61/sqrt(7)

= (50793.29, 66738.71)

(c) Here the darwin estimates are lesser than the 99% confidence interval so we can say the estimate are not consistenet with the data and we can say that there are more than 50000 worms per acre.

Question 4

Here as we see that all of the data contain 99% confidence interval, but that 99% confidance interval means that if we take sample size of 7 repeatedly then 99% of time we get the population mean under the 99% confidence interval. So if we take the sample with 6000 sample size, the confidence interval would be more confident and smaller. Here if we will substitute 6000 instead of 7, confidence interval reduced to way naroower.

99% CI (n = 6000)= y +- tdF,0.01 (s/sqrt(n)

= 58766 +- 3.7074 * 5689.61/sqrt(6000)

= (58439.68,59038.32)

(5) Here the 95% confidence interval for women =    y +- tdF,0.05 (s/n)

= 64 +- t0.05,5 * (3/6)

= 64 +- 2.5706 * (3/6) = (60.852, 67.148)

Here the 95% confidence interval for men =    y +- tdF,0.05 (s/n)

= 67 +- t0.05,5 * (3/6)

= 67 +- 2.5706 * (3/6) = (63.852, 70.148)

so yes the two confidence interval overlap.

Now we will repeat for n = 123

Here the 95% confidence interval for women =    y +- tdF,0.05 (s/n)

= 64 +- t0.05,5 * (3/123)

= 64 +- 2.5706 * (3/123) = (63.305, 64.695)

Here the 95% confidence interval for men =    y +- tdF,0.05 (s/n)

= 67 +- t0.05,5 * (3/123)

= 67 +- 2.5706 * (3/123) = (66.305, 67.695)

Here the confidence interval doesn't overlap.

(c) Here as we increase sample size, the confidence interval get reduced. So here sample size is important to find difference as that shows that the statistically signifance between any two groups. Increasing sample sizes reduce the margin of error.

Related Questions

Navigate

• Browse (All)
• Browse Y
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.