The mean annual income for people in a certain city (in thousands of dollars) is 43, with a standard deviation of 29. a pollster draws a sample of 41 people to interview. what is the probability that the sample mean income is less than 42 (thousands of dollars)?
a) 0.4721 b) 0.4129 c) 0.3483 d) 0.5279

There is a 41.29% probability that the sample mean income is less than 42 (thousands of dollars).

Step-by-step explanation:

The Central Limit Theorem estabilishes that, for a random variable X, with mean and standard deviation , a large sample size can be approximated to a normal distribution with mean and standard deviation

Problems of normally distributed samples can be solved using the z-score formula.

In a set with mean and standard deviation , the zscore of a measure X is given by:

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value 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. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this problem, we have that:

The mean annual income for people in a certain city (in thousands of dollars) is 43, with a standard deviation of 29. This means that .

A pollster draws a sample of 41 people to interview. This means that .

What is the probability that the sample mean income is less than 42 (thousands of dollars)?

This probability is the pvalue of Z when .

Due to the Central Limit Theorem, we use s instead of in the Zscore formula. So

has a pvalue of 0.4129.

This means that there is a 41.29% probability that the sample mean income is less than 42 (thousands of dollars).

step-by-step explanation: