Question

1 mr. gomez asked the 30 students in his class to each report how many star wars movies they have watched. the mean number of star wars movies watched was 5.25 with a standard deviation of 2.5 movies.
a. johann has watched four star wars movies. calculate the z - score for johann.
z=
b. interpret the z - score for johann in context.
c. sadie has a z - score of 1.1. how many star wars movies has she watched?
1.1 = \frac{x - 5.25}{2.5}
2 a grammy is a music award given to musicians, songwriters, producers, engineers, and industry professionals. the dotplot shows the ages, in years, of the ten oldest 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 formula for the z - score is $z=\frac{x-\mu}{\sigma}$, where $x$ is the individual value, $\mu$ is the mean, and $\sigma$ is the standard deviation.

Step2: Calculate z - score for Johann (Question 1a)

Given $\mu = 5.25$, $\sigma=2.5$, and $x = 4$. Substitute into the formula: $z=\frac{4 - 5.25}{2.5}=\frac{-1.25}{2.5}=- 0.5$

Step3: Interpret Johann's z - score (Question 1b)

Johann's z - score of $-0.5$ means that the number of Star - Wars movies he watched is 0.5 standard deviations below the mean number of movies watched by the students in the class.

Step4: Solve for Sadie's number of movies (Question 1c)

We have the equation $1.1=\frac{x - 5.25}{2.5}$. Cross - multiply: $1.1\times2.5=x - 5.25$. Then $2.75=x - 5.25$. Add 5.25 to both sides: $x=2.75 + 5.25=8$

Step5: Calculate percentile for Jimmy Carter (Question 2a)

Count the number of data points less than 94 in the dot - plot. There are 8 data points less than 94 out of 10. The percentile is $\frac{8}{10}\times100 = 80$th percentile. This means that 80% of the ten oldest Grammy winners are younger than Jimmy Carter when they won a Grammy.

Step6: Calculate z - score for Jimmy Carter (Question 2b)

Using the z - score formula $z=\frac{x-\mu}{\sigma}$ with $\mu = 89.4$ and $\sigma = 6.552$ and $x = 94$. $z=\frac{94 - 89.4}{6.552}=\frac{4.6}{6.552}\approx0.70$

Step7: Compare z - scores of Tony Bennett and Jimmy Carter (Question 2c)

Tony Bennett is 95 years old and Jimmy Carter is 94 years old. Since 95>94 and the mean and standard deviation are fixed for this data set, Tony Bennett's age is further above the mean than Jimmy Carter's age. So, Tony Bennett's z - score is greater than Jimmy Carter's z - score.

Answer:

1a. $z=-0.5$
1b. Johann's number of movies watched is 0.5 standard deviations below the class mean.
1c. Sadie has watched 8 Star - Wars movies.
2a. The percentile for Jimmy Carter's age is the 80th percentile. This means 80% of the ten oldest Grammy winners are younger than Jimmy Carter when they won a Grammy.
2b. The z - score for Jimmy Carter is approximately $0.70$.
2c. Tony Bennett's z - score is greater than Jimmy Carter's z - score because Tony Bennett's age (95 years) is greater than Jimmy Carter's age (94 years), and they are from the same data set with a fixed mean and standard deviation.