Question

a grammy is a music award given to musicians, songwriters, producers, engineers, and industry professionals. the dot - plot shows the ages, in years, of the past grammy winners. the distribution of ages has a mean of 89.4 years and a standard deviation of 6.552 years.
a. jimmy carter was 94 years old when he won a grammy. find and interpret the percentile for this age.
b. calculate and interpret the z - score for jimmy carter. round to the nearest hundredth.
c. tony bennett was 95 years old when he last won a grammy award. without doing any calculations, compare the z - score of tony bennetts age to the z - score of jimmy carter. explain your reasoning.

Explanation:

Step1: Recall z - score formula

The z - score formula is $z=\frac{x - \mu}{\sigma}$, where $x$ is the data value, $\mu$ is the mean, and $\sigma$ is the standard deviation. Given $\mu = 89.4$ and $\sigma=6.552$.

Step2: Calculate z - score for Jimmy Carter

For Jimmy Carter, $x = 94$. Substitute into the formula: $z=\frac{94 - 89.4}{6.552}=\frac{4.6}{6.552}\approx0.70$. This means Jimmy Carter's age is approximately $0.70$ standard - deviations above the mean age of Grammy winners.

Step3: Calculate z - score for Tony Bennett

For Tony Bennett, $x = 95$. Then $z=\frac{95 - 89.4}{6.552}=\frac{5.6}{6.552}\approx0.85$.

Step4: Compare z - scores

Since $0.85>0.70$, Tony Bennett's age is relatively higher compared to the mean age of Grammy winners than Jimmy Carter's age. The z - score represents the number of standard deviations a data point is from the mean. A higher z - score indicates that the data point is further above the mean in a positive direction.

Answer:

The z - score for Jimmy Carter is approximately $0.70$, which means his age is about $0.70$ standard deviations above the mean age of Grammy winners. The z - score for Tony Bennett is approximately $0.85$, meaning his age is about $0.85$ standard deviations above the mean age of Grammy winners. Tony Bennett's age is relatively higher compared to the mean age of Grammy winners than Jimmy Carter's age because $0.85 > 0.70$.