Question

Suppose a research firm conducted a survey to determine the average amount of money steady smokers spend on cigarettes during a week. The sample size is 100. The population mean is $20 and the population standard deviation is $5. What is the probability that a sample of 100 steady smokers spend within $1 of the mean?

Answer 1

The probability that a sample of 100 steady smokers spend within $1 of the mean is calculated using the standard error and z-scores, referencing a z-table to find the corresponding probabilities.

Step-by-step explanation:

To calculate the probability that a sample of 100 steady smokers spend within $1 of the mean ($20), we use the concept of the standard error (SE) and the z-score in a normal distribution.

The standard error is calculated by dividing the population standard deviation by the square root of the sample size (n).

SE = σ / √n
SE = $5 / √100
SE = $5 / 10
SE = $0.50

Next, we determine the z-scores for $1 above and $1 below the mean.

Z = (X - μ) / SE
Z for $21 = ($21 - $20) / $0.50 = 2
Z for $19 = ($19 - $20) / $0.50 = -2

Using the z-table, we find the probabilities corresponding to these z-scores and subtract the smaller from the larger to find the probability that the sample mean is within $1 of the population mean.