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

Answer 1

Complete Question:

a) Is it plausible that X is normally distributed?

b) For a random sample of 50 such pairs, what is the (approximate) probability that the sample mean courtship time is between 100 min and 125 min?

Answer:

a) It is plausible that X is normally distributed

b) probability that the sample mean courtship time is between 100 min and 125 min is 0.5269

Explanation:

a)X denotes the courtship time for the scorpion flies which indicates that is a real - valued random variable, and since normal distribution is a continuous probability distribution for a real valued random variable, it is plausible that X is normally distributed.

b) Probability that the sample mean courtship time is between 100 min and 125 min

`\mu = 120\\n = 50`

`P(x_(1) < \bar{X} < x_(2) ) = P(z_(2) < (x_(2)- \mu )/(SD) ) - P(z_(1) < (x_(2)- \mu )/(SD))`

`SD = \sqrt{(\sigma^(2) )/(n) } \\SD = \sqrt{(110^(2) )/(50) } \\SD = 15.56`

`P(100 < \bar{X} <125 ) = P(z_(2) < (125- 120 )/(15.56) ) - P(z_(1) < (100- 120 )/(15.56))\\P(100 < \bar{X} <125 ) = P(z_(2) < 0.32 ) - P(z_(1) < -1.29)`

From the probability distribution table:

`P(z_(2) < 0.32 ) = 0.6255\\ P(z_(1) < -1.29) = 0.0986`

`P(100 < \bar{X} <125 ) = 0.6255 - 0.0986\\P(100 < \bar{X} <125 ) =0.5269`