Scientists researched the emerging rate from pupa of monarch butterfly (Danaus p

Question

Scientists researched the emerging rate from pupa of monarch butterfly (Danaus plexippus) in Northern Florida. Many pupas of monarch died because of parasitoids (parasite insects, like wasps). They observed 1,000 pupas in the field, and 268 butterflies emerged from pupa (success), but 732 were dead (failure). We assume each pupa is independent and identical, so those survival processes can follow binomial distribution. Calculate the sample proportion of survived pupas, p cap. What is the best point estimator of the TRUE "p"? Calculate the sample proportion of dead pupas, q cap. What is the theoretical standard deviation of p cap? Assume each pupa's survival is a Bernoulli experiment. What is the standard error for p cap, sigma_p cap? What is the critical value Z_90 when the level of confidence is 90%? What is the margin of error of p, when the level of confidence is 90%? What is the 90% Confidence Intervals of p? Interpret (h). You want your estimate must be accurate within 0.5% (E = 0.005) of the true survival rate, p when the level of confidence is 90%. What is the minimum required sample size, n? Use p cap and q cap to calculate.

Explanation / Answer

Question 2 .

Total pupas in the field = 1000

Emerged pupas (success) = 268

Dead pupas (Failure) = 732

(a) sample proportion of survived "pupa" p^ = 268/1000= 0.268

(b) The best point estimate of the TRUE "p" = p^ = 0.268

(c) sample proortion of Dead "pupa" q^ = 732/1000 = 0.732

(d) Theoretical standard deviation of p^ = sqrt [ pq * N] = sqrt [0.268 * 0.732 / 1000] = 0.014

(e) Standard error of p^ = sqrt [ p^q^ * N] = sqrt [0.268 * 0.732 / 1000] = 0.014

(f) Critical value Z0.90 when level of confidence is 90%

Z0.90 = 1.645

(g) Margin of error of p when level of confidence is 90% .

Margin of error = Critical test statistic * Standard eror of sample

= z0.10 * 0.014 = 1.645 * 0.014 = 0.023

(h) 90% confidence interval for p = p^ +- Z0.90 sqrt [pq/n] = 0.268 +- 0.023 = (0.245, 0.291)

(i) We can interpret from h that there are 95 % probability that percentage survival for butterflies from pupa is in between 24.5 % to 29.1 %.

(j) Let say true survival rate is p so margin of error will 0.005p

so z0.90 * se0 < 0.005 p

where se0 = sqrt [p^q^/N]

and Z0.90 = 1.645

1.645 * sqrt [0.268 * 0.732/N] < 0.005 * 0.268

N = 295643

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