Answer:
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).