Question

in a recent study on world happiness, participants were asked to evaluate their current lives on a decimal scale from 0 to 10, where 0 represents the worst possible life and 10 represents the best possible life. the mean response was 5.3 with a standard deviation of 2.4. assume the distribution can be approximated by a normal distribution. (a) what response represents the 91st percentile? (b) what response represents the 61st percentile? (c) what response represents the first quartile? (a) the response that represents the 91st percentile is . (round to two decimal places as needed.)

Explanation:

Step1: Recall z - score formula

For a normal distribution $X\sim N(\mu,\sigma^{2})$, the z - score is given by $z=\frac{x - \mu}{\sigma}$, and we want to find $x$ given a percentile. We first find the z - score corresponding to the percentile from the standard normal distribution table, and then solve for $x$. The formula for $x$ is $x=\mu+z\sigma$. Here, $\mu = 5.3$ and $\sigma=2.4$.

Step2: Find z - score for 91st percentile

Looking up the 91st percentile in the standard - normal distribution table (the area to the left of the z - value is 0.91). The z - score $z_{0.91}\approx1.34$.

Step3: Calculate the value of x

Substitute $\mu = 5.3$, $\sigma = 2.4$, and $z = 1.34$ into the formula $x=\mu+z\sigma$.
$x=5.3+1.34\times2.4$
$x=5.3 + 3.216$
$x = 8.52$

Step4: Find z - score for 61st percentile

Looking up the 61st percentile in the standard - normal distribution table (the area to the left of the z - value is 0.61). The z - score $z_{0.61}\approx0.28$.

Step5: Calculate the value of x for 61st percentile

Substitute $\mu = 5.3$, $\sigma = 2.4$, and $z = 0.28$ into the formula $x=\mu+z\sigma$.
$x=5.3+0.28\times2.4$
$x=5.3+0.672$
$x = 5.97$

Step6: Find z - score for first quartile

The first quartile is the 25th percentile. Looking up the 25th percentile in the standard - normal distribution table (the area to the left of the z - value is 0.25). The z - score $z_{0.25}\approx - 0.67$.

Step7: Calculate the value of x for first quartile

Substitute $\mu = 5.3$, $\sigma = 2.4$, and $z=-0.67$ into the formula $x=\mu+z\sigma$.
$x=5.3+( - 0.67)\times2.4$
$x=5.3-1.608$
$x = 3.69$

Answer:

(a) 8.52
(b) 5.97
(c) 3.69