The average cost of a one-bedroom apartment in a town is $650 per month. What is the probability of randomly selecting a sample of 50 one-bedroom apartments in this town and getting a sample mean of less than $630 if the population standard deviation is $100?

Probability of randomly selecting a sample of 50 one-bedroom apartments in this town and getting a sample mean of less than $630 is 0.0793.

Step-by-step explanation:

We are given that the average cost of a one-bedroom apartment in a town is $650 per month. Assume the population standard deviation is $100.

Also, a sample of 50 one-bedroom apartments is selected.

Let = sample mean cost

The z-score probability distribution for sample mean is given by ;

                Z = ~ N(0,1)

where, = population average cost = $650

            = population standard deviation = $100

            n = sample of one-bedroom apartments = 50

The Z-score measures how many standard deviations the measure is away from the mean. After finding the Z-score, we look at the z-score table and find the p-value (area) associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X.  

Now, the probability that the sample mean is less than $630 is given by = P( < $630)

   P( < $630) = P( < ) = P(Z < -1.41) = 1 - P(Z 1.41)

                                                        = 1 - 0.9207 = 0.0793

The above probability is calculated using z table by looking at value of x = 1.41 in the z table which have an area of 0.92073.

Therefore, probability that the sample mean is less than $630 is 0.0793.

We can use standard normal variate and use z tables to find the needed probability.

The needed probability is 0.0793

Let the sample mean be tracked by random variable

Since standard deviation is known already, we can transform the r.v. approximately into standard normal variate by:

Then we will have:

The value 0.9207 is p value for z-score 1.41 seen from the z-score table.

Thus, the needed probability is 0.0793

Learn more about z-score here: