It is known that 65% of all brand A external hard drives work in a satisfactory manner throughout the warranty period (are "successes"). Suppose that n = 15 drives are randomly selected. Let X = the number of successes in the sample. The statistic X/n is the sample proportion (fraction) of successes. Obtain the sampling distribution of this statistic. [Hint: One possible value X/n is 0.2, corresponding to X = 3. What is the probability of this value (what kind of random variable is X)?] (Round your answers to three decimal places.)

This is a discrete variable, looks like it could have a binomial distribution, to be certain you have to first check if it follows the criteria of a binomial distribution:

Binomial criteria:

1. The number of observation of the trial is fixed (In this case n = 15)

2. Each observation in the trial is independent, this means that none of the trials will affect the probability of the next trial (In this case, Each external hard drive observed does not affect on the probability of the next one being defective throughout the warranty period)

3. The probability of success in the same from one trial to another (In this case our "success" will be an external hard drive that works satisfactorily throughout the warranty period; ρ=0,65)

So X≈ Bi (n;ρ)

Where n represents the sample (n=15) and ρ is the probability of success (ρ=0.65)

The sample proportion has E(^ρ)= ρ and V(^ρ)= (ρ(1-ρ))/n

Its distribution(for a big enough n) is ^ρ≈N (ρ; (ρ(1-ρ))/n)

Under the distribution of the sample proportion, the probability of X=3 is cero, since the Normal distribution is continuous and the probability of getting exactly a value is 0

Note: if you want to calculate it under the binomial distribution:

P(X=3) = P(X≤3) - P(X≤2) = 0 - 0 = 0

I hope this helps!