Question

We have these hypotheses:

H0: the amount of active ingredient in a pharmaceutical pill is 5 mg

H1: the amount is below 5 mg

We can take a random sample of 40 such pills and find the amount of active ingredient in them, and let the sample average be x-bar. We also know that the standard deviation of the amount of active ingredient is 0.3 mg.

If H0 is right, what is the (approximate) distribution of ( x-bar minus 5) / (0.3 / square root of 40 ) ) ?

		A. Normal with mean zero, standard deviation 1
		B. Normal with mean 5, standard deviation 0.3
		C. Normal with mean 5, standard deviation (0.3 / square root of 40)
		D. binomial with n = 40, p = 0.3

Answer 1

Answer: option A) Normal with mean 0, standard deviation 1

Since H0: u_{​​​​​​0}​​​​​=5 versus H1: u<5

And standard deviation is known then

According to given data

It follows z which is approximately equal to normal distribution with mean 0 and standard deviation 1 due to Central limit theorem. Where Central limit theorem states that mean of sample observations (x-bar) is approximately equal to population mean, when sample is large and variance of population is known .

Where z =((x-bar)- mean=5)/(0.3/√40)) (test statistics)