Question

statistics - chapter one and two - tam 2023
for questions 1-5, identify the whole term being described. you may reuse terms from
the vocabulary list. (1 point each)

1. population (1)
2. sample
3. qualitative data
4. random sample
5. biased data
6. why are samples often used in statistical surveys instead of a population?
7. if a sample is used, why is it so important that it be representative?
8. is it possible for 60% of the numbers in a data set to be “above average”? explain.
9. what type of graph(s) are appropriate for qualitative data? for quantitative data?

for the given data set, find each of the following (use calculator or by hand):
55, 69, 75, 76, 78, 80, 81, 81, 89
10 mean
11 standard deviation
12 five - number summary
13 determine if there are outliers
14 construct a box plot

Explanation:

10. Mean

Step 1: Sum the data points

The data set is \( 55, 69, 75, 76, 78, 80, 81, 81, 89 \).
Sum \( = 55 + 69 + 75 + 76 + 78 + 80 + 81 + 81 + 89 \)
\( = 55 + 69 = 124 \); \( 124 + 75 = 199 \); \( 199 + 76 = 275 \); \( 275 + 78 = 353 \); \( 353 + 80 = 433 \); \( 433 + 81 = 514 \); \( 514 + 81 = 595 \); \( 595 + 89 = 684 \).

Step 2: Divide by number of data points (\( n = 9 \))

Mean \( = \frac{\text{Sum}}{n} = \frac{684}{9} = 76 \).

11. Standard Deviation

Step 1: Find the mean (already calculated: \( \mu = 76 \))

Step 2: Calculate squared deviations from the mean

For each data point \( x \), compute \( (x - \mu)^2 \):

• \( (55 - 76)^2 = (-21)^2 = 441 \)
• \( (69 - 76)^2 = (-7)^2 = 49 \)
• \( (75 - 76)^2 = (-1)^2 = 1 \)
• \( (76 - 76)^2 = 0^2 = 0 \)
• \( (78 - 76)^2 = 2^2 = 4 \)
• \( (80 - 76)^2 = 4^2 = 16 \)
• \( (81 - 76)^2 = 5^2 = 25 \)
• \( (81 - 76)^2 = 5^2 = 25 \)
• \( (89 - 76)^2 = 13^2 = 169 \)

Step 3: Sum the squared deviations

Sum \( = 441 + 49 + 1 + 0 + 4 + 16 + 25 + 25 + 169 = 730 \)

Step 4: Compute variance (population variance, since it’s a data set, not a sample)

Variance \( \sigma^2 = \frac{\text{Sum of squared deviations}}{n} = \frac{730}{9} \approx 81.11 \)

Step 5: Take the square root for standard deviation

Standard Deviation \( \sigma = \sqrt{\frac{730}{9}} \approx \sqrt{81.11} \approx 9.01 \)

12. Five - number Summary

Step 1: Order the data (already ordered: \( 55, 69, 75, 76, 78, 80, 81, 81, 89 \))

Step 2: Find Minimum (\( \text{Min} \))

\( \text{Min} = 55 \)

Step 3: Find Maximum (\( \text{Max} \))

\( \text{Max} = 89 \)

Step 4: Find Median (middle value, \( n = 9 \), so 5th term)

Median \( = 78 \)

Step 5: Find Q1 (median of lower half: \( 55, 69, 75, 76 \))

Lower half has 4 terms, median is average of 2nd and 3rd: \( \frac{69 + 75}{2} = \frac{144}{2} = 72 \)

Step 6: Find Q3 (median of upper half: \( 80, 81, 81, 89 \))

Upper half has 4 terms, median is average of 2nd and 3rd: \( \frac{81 + 81}{2} = 81 \)

Five - number summary: \( 55, 72, 78, 81, 89 \)

13. Determine if there are outliers

Answer:

s:

1. Mean: \( \boldsymbol{76} \)
2. Standard Deviation: \( \boldsymbol{\approx 9.01} \) (or exact \( \sqrt{\frac{730}{9}} \))
3. Five - number Summary: \( \boldsymbol{55, 72, 78, 81, 89} \)
4. Outlier: \( \boldsymbol{55} \) (is an outlier)
5. Box Plot: (As described: box from 72 to 81, median at 78, lower whisker to 69, upper whisker to 89, outlier at 55)