Respuesta :
Answer:
Correct option: (d) 75.4 days.
Step-by-step explanation:
Let X = number of days that homes stay on the market before selling.
The average number of days that homes stay on the market before selling is, μ = 78.4 days.
The standard deviation of the number of days that homes stay on the market before selling is, σ = 11 days.
A sample of size, n = 36 homes are selected from the multiple listing service.
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=78.4[/tex]
And the standard deviation of the distribution of sample mean is given by,
[tex]\sigma_{\bar x}=\frac{\sigma}{\sqrt{n}}=\frac{11}{\sqrt{36}}=1.83[/tex]
So, the distribution of sample mean is, [tex]\bar X\sim N(78.4,\ 1.83^{2})[/tex].
Let [tex]\bar x[/tex] be the value of the sample mean above which 95% of all possible sample means fall.
That is,
[tex]P(\bar X>\bar x)=0.95\\P(\bar X<\bar x)=0.05\\P(Z<z)=0.05[/tex]
The value of z for the above probability is:
z = -1.645.
*Use a z-table for the value.
Compute the value of [tex]\bar x[/tex] as follows:
[tex]z=\frac{\bar x-\mu_{\bar x}}{\sigma_{\bar x}}\\-1.645=\frac{\bar x-78.4}{1.83}\\\bar x=78.4-(1.645\times 1.83)\\\bar x=75.389\\\bar x\approx75.4[/tex]
Thus, the sample mean above which 95% of all possible sample means fall is 75.4 days.
