Question

4. 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 this article). (a) Is it plausible that X is normally distributed

Answer 1

On this case if we analyze the situation we see that we are analyzing a random variable related to the time.

And the time cannot be negative, and the standard deviation is almost equal to the mean, too large. So will not plausible to say that the random variable X would be distributed normal.

Explanation:

The empirical rule, also known as three-sigma rule or 68-95-99.7 rule, "is a statistical rule which states that for a normal distribution, almost all data falls within three standard deviations (denoted by σ) of the mean (denoted by µ)".

Let X the random variable who represent the courtship time (minutes).

From the problem we have the mean and the standard deviation for the random variable X.
`E(X)=120, Sd(X)=110`

So we can assume
`\mu=120 , \sigma=110`

On this case in order to check if the random variable X follows a normal distribution we can use the empirical rule that states the following:

• The probability of obtain values within one deviation from the mean is 0.68

• The probability of obtain values within two deviation's from the mean is 0.95

• The probability of obtain values within three deviation's from the mean is 0.997

So we need values such that

`P(X<\mu -\sigma)=P(X <10)=0.16`

`P(X>\mu +\sigma)=P(X >230)=0.16`

`P(X<\mu -2*\sigma)P(X<-100)=0.025`

`P(X>\mu +2*\sigma)P(X>340)=0.025`

`P(X<\mu -3*\sigma)=P(X<-210)=0.0015`

`P(X>\mu +3*\sigma)=P(X>450)=0.0015`

On this case if we analyze the situation we see that we are analyzing a random variable related to the time.

And the time cannot be negative, and the standard deviation is almost equal to the mean, too large. So will not plausible to say that the random variable X would be distributed normal.