Thank you for visiting Suppose the ages of people who own their homes are normally distributed with a mean of 42 years and a standard deviation of 3 2. This page is designed to guide you through key points and clear explanations related to the topic at hand. We aim to make your learning experience smooth, insightful, and informative. Dive in and discover the answers you're looking for!
Answer :
The ages of people who own their homes are normally distributed with a mean of 42 and a standard deviation of 3.2. To find the age at which 75% of the home owners are older than this age, we can use the standard normal distribution table to find the z-score, which is approximately 0.674. The correct option is 44.2, as 75% of the home owners are older than 44.2.
Suppose ages of people who own their homes are normally distributed with a mean of 42 years and a standard deviation of 3.2 years. We need to find the age at which 75% of the home owners are older than this age.How to find the age?We can use the standard normal distribution table. The standard normal distribution is a normal distribution with a mean of 0 and standard deviation of 1. Using this table, we can find the z-score which corresponds to the area to the left of a particular value.Suppose z is the z-score such that the area to the left of z is 0.75. This means that 75% of the distribution is to the left of z. We can use this z-score to find the age at which 75% of the home owners are older than this age.What is the formula for z-score?The formula for finding the z-score for a value x in a normal distribution with mean μ and standard deviation σ is as follows
[tex]:$$z=\frac{x-\mu}{\sigma}$$[/tex]
We can rearrange this formula to find the value of x. Here's how:$$x=\sigma z + \mu$$
Now, let's find the z-score which corresponds to the area to the left of 0.75. Using a standard normal distribution table
we can find that this z-score is approximately 0.674.Using the above formula, we can find the age x at which 75% of the home owners are older than this age. We know that μ = 42 and σ = 3.2. Therefore, we have:$$x = 3.2 \times 0.674 + 42 = 44.16$$
Therefore, approximately 75% of the home owners are older than 44.16. So, the correct option is 44.2.Approximately 75% of the home owners are older than 44.2.
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