Respuesta :
Answer:
The probability that the sample mean will be within 0.5 of the population mean is 0.3328.
Step-by-step explanation:
It is provided that a random variable X has mean, μ = 50 and standard deviation, σ = 7.
A random sample of size, n = 36 is selected.
According to the Central Limit Theorem if we have a population with mean μ and standard deviation σ and we take appropriately huge random samples (n ≥ 30) from the population with replacement, then the distribution of the sample mean will be approximately normally distributed.
Then, the mean of the distribution of sample mean is given by,
[tex]\mu_{\bar x}=\mu=50[/tex]
And the standard deviation of the distribution of sample mean is given by,
[tex]\sigma_{\bar x}=\frac{\sigma}{\sqrt{n}}=\frac{7}{\sqrt{36}}=1.167[/tex]
So, the distribution of the sample mean of X is N (50, 1.167²).
Compute the probability that the sample mean will be within 0.5 of the population mean as follows:
[tex]P(|\bar X-\mu_{\bar x}|\leq 0.50)=P(-0.50<(\bar X-\mu_{\bar x})<0.50)[/tex]
[tex]=P(\frac{-0.50}{1167}<\frac{\bar X-\mu_{\bar x}}{\sigma_{\bar x}}<\frac{0.50}{1.167})\\\\=P(-0.43<Z<0.43)\\\\=P(Z<0.43)-P(Z<-0.43)\\\\=0.66640-0.33360\\\\=0.3328[/tex]
Thus, the probability that the sample mean will be within 0.5 of the population mean is 0.3328.
