Final answer:
Approximately 11.12% of day care costs are more than $9830 annually. About 58.03% of the costs fall between $7250 and $9830. In a sample of 120 families, approximately 13 pay more than $9830 annually for day care.
Step-by-step explanation:
Calculating Percentages in a Normal Distribution
To find what percent of day care costs are more than $9830 annually, we need to calculate the z-score, and then determine the corresponding percentile from the standard normal distribution table. The z-score is calculated as follows:
Z = (X - μ) / σ
Where X is $9830, μ (mu) is the mean of $8000, and σ (sigma) is the standard deviation of $1500. Substituting the values in:
Z = ($9830 - $8000) / $1500
Z = $1830 / $1500
Z ≈ 1.22
Using the standard normal distribution table, we find that the area to the right of Z = 1.22 is approximately 0.1112, which represents the percent of costs that are more than $9830. Thus, about 11.12% of day care costs are more than $9830 annually.
To find what percentage of day care costs are between $7250 and $9830, we will calculate the z-scores for both values:
Z for $7250 = ($7250 - $8000) / $1500 = -0.50
Z for $9830, as calculated before, is 1.22.
The area to the left of Z = -0.50 is approximately 0.3085 and to the left of Z = 1.22 is approximately 0.8888. So the area between them is 0.8888 - 0.3085 = 0.5803, which means that 58.03% of the costs fall between $7250 and $9830 annually.
To find the number of families that pay more than $9830 annually for day care, we use the proportion derived earlier (0.1112) and multiply this by the total sample size of 120 families:
Number of families = 0.1112 * 120 = 13.344
Since we cannot have a fraction of a family, we round this to 13 families.