Question

Let X denote the courtship time for a randomly selected female-male pair of mating scorpion flies (time from the beginning of interaction until mating). Suppose the mean value of X is 120 min and the standard deviation of X is 110 min (suggested by data in the article "Should I Stay or Should I Go? Condition- and Status-Dependent Courtship Decisions in the Scorpion Fly Panorpa Cognate"+). (a) Is it plausible that X is normally distributed? No, courtship time cannot plausibly be normally distributed. Since X must be non-negative, realistically the interval u = 30 should be entirely non-negative which is not true in this case. Yes, courtship time is plausibly normally distributed since the mean of X is larger than 100. No, courtship time cannot plausibly be normally distributed. Since X must be negative, realistically the interval u + 30 should be entirely negative which is not true in this case. Yes, courtship time is plausibly normally distributed. Since X must be negative, realistically the interval u 30 should be entirely negative which is true in this case. Yes, courtship time is plausibly normally distributed. Since X must be non-negative, realistically the interval u = 30 should be entirely non-negative which is true in this case. (b) For a random sample of 50 such pairs, what is the (approximate) probability that the sample mean courtship time is between 115 min and 135 min? (Round your answer to four decimal places.) (c) For a random sample of 50 such pairs, what is the (approximate) probability that the total courtship time exceeds 150 min? (Round your answer to four decimal places.) (d) Could the probability requested in (b) be calculated from the given information if the sample size were 15 rather than 50? Explain. Yes. According to the guidelines n should be less than 30 in order to apply the central limit theorem. No. According to the guidelines n should be greater than 30 in order to apply the central limit theorem.

Answer 1

b)

for normal distribution z score =(X-μ)/σx
here mean=       μ=	120
std deviation   =σ=	110.0
sample size       =n=	50
std error=σx̅=σ/√n=	15.5563

probability =

P(115<X<135)

=

P(-0.32<Z<0.96)=

0.8315-0.3745=

0.4570

c)

probability =

P(X>150)

=

P(Z>1.93)=

1-P(Z<1.93)=

1-0.9732=

0.0268

d)

No. According to the guidelines n should be greater than 30 in order to apply the central limit theorem.