Question

1. a statistics class wanted to examine the texting habits of their classmates. so, they took a random sample of 15 kids at their school and counted how many texts they had sent in the last 6 hours. the data are shown below.

127, 44, 28, 83, 0, 6, 70, 6, 5, 213, 73, 20, 214, 28, 11
create a boxplot of the given data. be sure to indicate outliers and include proper labels.

1. two different golfers play at different courses. adam shot 67 on course a, which has a mean score of 72.1 and a standard deviation of 3.3. rachel shot 69 on course b, with a mean score of 72.5 and a standard deviation of 1.3. use z-scores to determine whose score was more impressive? (remember, a lower score is better in golf)

1. (referring to problem 11) why would it be preferred to use z-scores to compare these players’ scores rather than just compare their actual scores?

Explanation:

Question 10

Step 1: Organize the data

First, we need to sort the data in ascending order. The given data is: 127, 44, 28, 83, 0, 6, 70, 6, 5, 213, 73, 20, 214, 28, 11. After sorting: 0, 5, 6, 6, 11, 20, 28, 28, 44, 70, 73, 83, 127, 213, 214.

Step 2: Find the five - number summary

• Minimum value ($\text{Min}$): The smallest value in the sorted data, which is 0.
• First Quartile ($Q_1$): The median of the lower half of the data. The lower half of the data (excluding the median if $n$ is odd) for $n = 15$ (odd) is the first 7 values: 0, 5, 6, 6, 11, 20, 28. The median of these 7 values is the 4th value, so $Q_1=6$.
• Median ($\text{Med}$ or $Q_2$): The middle value of the sorted data. For $n = 15$, the median is the 8th value, so $\text{Med}=28$.
• Third Quartile ($Q_3$): The median of the upper half of the data. The upper half of the data (excluding the median) is the last 7 values: 44, 70, 73, 83, 127, 213, 214. The median of these 7 values is the 4th value, so $Q_3 = 83$.
• Maximum value ($\text{Max}$): The largest value in the sorted data, which is 214.

Step 3: Calculate the inter - quartile range (IQR)

The inter - quartile range is calculated as $IQR=Q_3 - Q_1$. So, $IQR = 83-6=77$.

Step 4: Identify outliers

We use the formula for outliers:

• Lower bound for outliers: $Q_1-1.5\times IQR=6 - 1.5\times77=6 - 115.5=- 109.5$
• Upper bound for outliers: $Q_3 + 1.5\times IQR=83+1.5\times77=83 + 115.5 = 198.5$

Any data point less than the lower bound or greater than the upper bound is an outlier. Looking at the data, 213 and 214 are greater than 198.5, so 213 and 214 are outliers.

Step 5: Create the boxplot

• Draw a number line that covers the range of the data from 0 to 214.
• Draw a box from $Q_1 = 6$ to $Q_3=83$. Inside the box, draw a line at the median $\text{Med}=28$.
• Draw whiskers from the box to the minimum value (0) and to the maximum non - outlier value (127).
• Plot the outliers (213 and 214) as individual points above the upper whisker.
• Label the boxplot with the title "Boxplot of Text Messages Sent in 6 Hours" and label the axes (e.g., "Number of Text Messages" on the y - axis and "Data Points" on the x - axis, or a simple number line with appropriate labels).

Question 11

Step 1: Recall the z - score formula

The z - score formula is $z=\frac{x-\mu}{\sigma}$, where $x$ is the data point, $\mu$ is the mean, and $\sigma$ is the standard deviation. A lower z - score (since lower golf scores are better) indicates a more impressive performance relative to the course.

Step 2: Calculate Adam's z - score

For Adam: $x = 67$, $\mu=72.1$, $\sigma = 3.3$.
$z_{Adam}=\frac{67 - 72.1}{3.3}=\frac{- 5.1}{3.3}\approx - 1.55$

Step 3: Calculate Rachel's z - score

For Rachel: $x = 69$, $\mu = 72.5$, $\sigma=1.3$.
$z_{Rachel}=\frac{69 - 72.5}{1.3}=\frac{-3.5}{1.3}\approx - 2.69$

Step 4: Compare the z - scores

Since $-2.69<-1.55$, Rachel's z - score is lower. In golf, a lower score relative to the mean (indicated by a lower z - score) is more impressive.

Question 12

The two golfers are playing on different courses. Different courses have different mean scores ($\mu$) and different standard deviations ($\sigma$). Comparing the actual scores (67 and 69) directly does not take into account the difficulty of the courses (measured by the mean and standard deviation of scores on each course). Z - scores standardize the scores by measuring how many standard deviations a score is away from the mean of its respective distribution (course). This allows us to compare the relative performance of the two golfers, as it adjusts for the different centers (means) and spreads (standard deviations) of the score distributions on the two different courses.

Answer:

s:

Question 10

The boxplot is created with minimum = 0, $Q_1 = 6$, median = 28, $Q_3=83$, maximum (non - outlier)=127, and outliers at 213 and 214. (The boxplot is drawn as described in the steps above with appropriate labels).

Question 11

Rachel's score was more impressive because her z - score ($z\approx - 2.69$) is lower than Adam's z - score ($z\approx - 1.55$).

Question 12

It is preferred to use z - scores because the golfers play on different courses with different mean scores and standard deviations. Z - scores standardize the scores to account for the different centers and spreads of the score distributions on the two courses, allowing for a fair comparison of relative performance.