Question

Let the random variable X follow a normal distribution with a mean of μ and a standard deviation of σ. Let X ₁ be the mean of a sample of 36 observations randomly chosen from this population, and X ₂ be the mean of a sample of 25 observations randomly chosen from the same population.

a) How are X ₁ and X ₂ distributed? Write down the form of the density function and the corresponding parameters.

b) Evaluate the statement:

P(μ - 0.2σ <X ₁ < μ + 0.2σ) < P(μ - 0.2σ <X ₂ < μ + 0.2σ) as to whether it is true or false.

Answer 1

1)X1 follows normal distribution with mean and standard deviation of ​​​​​​

X2 follows normal distribution with mean and standard deviation of ​​​​​​

2)The statement is false because the probability is greater for X1 than for X2.

The first probability reduces to

P(-1.2<Z<1.2)=P(Z<1.2)-P(Z<-1.2)=P(Z<1.2)-{1-P(Z<1.2)}=0.7699

Similarly, the second one reduces to

P(-1<Z<1) and preceding the same way as before we get the probability as 0.6827