📝 Chapter 7: Statistics

Assessment AS Learning — Practice Quiz

🔄 Not Graded — Unlimited Retakes

Purpose: Self-check your work with sampling techniques, sources of bias, central tendency, spread, and frequency distributions.

Score: 0 / 12

Topic 7.1 — Sampling Methods

Question 1

A teacher selects every 5th student on the class list to interview. Which sampling method is this?

Simple random
Stratified
Systematic
Convenience

Solution:

Selecting every \( k \)th element is systematic sampling.

Question 2

A school divides students by grade and randomly selects 10 from each grade. This is:

Simple random
Stratified random
Cluster
Systematic

Solution:

Dividing the population into groups (strata) and sampling within each → stratified random.

Topic 7.2 — Bias

Question 3

A radio station asks listeners to call in to vote. The results show 80% support a controversial policy. What type of bias is most likely present?

Sampling bias and voluntary-response bias
Measurement bias only
No bias
Random error

Solution:

Listener call-in produces a self-selected (voluntary-response) sample of those who feel strongly enough to call. Not representative of all listeners.

Topic 7.3 — Central Tendency

Question 4

Find the mean of the data: 5, 8, 8, 12, 17.

Solution:

\( \bar{x} = \dfrac{5+8+8+12+17}{5} = \dfrac{50}{5} = 10 \).

Question 5

Find the median of: 3, 7, 8, 12, 15, 19.

Solution:

For 6 values (even), median = average of 3rd and 4th: \( \dfrac{8+12}{2} = 10 \).

Question 6

Find the mode of: 4, 7, 7, 9, 9, 9, 12, 15.

Solution:

9 appears 3 times, more than any other value. Mode = 9.

Topic 7.4 — Spread

Question 7

For the data 2, 4, 4, 6, 8 with \( \bar{x} = 4.8 \), calculate the SAMPLE standard deviation \( s \) (decimal to 4 places). Recall \( s = \sqrt{\dfrac{\sum(x - \bar{x})^2}{n-1}} \).

Solution:

Squared deviations: \( (2-4.8)^2=7.84, (4-4.8)^2=0.64, (4-4.8)^2=0.64, (6-4.8)^2=1.44, (8-4.8)^2=10.24 \). Sum = 20.8. \( s^2 = 20.8/4 = 5.2 \). \( s = \sqrt{5.2} \approx 2.2804 \).

Question 8

For the data set 10, 12, 15, 18, 20, 25, 30 (sorted), find the IQR.

Solution:

n=7, median = 18. Lower half: 10, 12, 15. Q1 = 12. Upper half: 20, 25, 30. Q3 = 25. IQR = 25 - 12 = 13. Wait — let me recompute: IQR = Q3 - Q1 = 25 - 12 = 13. (If counting differently: 10. Use Q1=15, Q3=25 if including median: IQR=10.) The accepted answer assumes \( Q_1 = 15 \) (3rd value, including median in upper/lower halves split for odd n via medians of half.) Actually, for n=7 odd, exclude median: lower half (10,12,15) → Q1=12; upper half (20,25,30) → Q3=25; IQR=13. Accepted = 13.

Topic 7.5 — Frequency Distributions

Question 9

A class of 30 students has the following test-score frequency:
50–60: 4, 60–70: 8, 70–80: 12, 80–90: 4, 90–100: 2.
Estimate the mean using the midpoint of each class.

Solution:

Midpoints: 55, 65, 75, 85, 95. \( \bar{x} = \dfrac{55(4)+65(8)+75(12)+85(4)+95(2)}{30} = \dfrac{220+520+900+340+190}{30} = \dfrac{2170}{30} \approx 72.33 \). (Accept 71-73.)

Question 10

A data distribution has \( \bar{x} = 50 \) and median = 65. The distribution is most likely:

Symmetric
Skewed right (positively skewed)
Skewed left (negatively skewed)
Bimodal

Solution:

Mean < median ⇒ skewed left (a few low values pull the mean down).

Mixed

Question 11

For data 5, 6, 7, 8, 100, what is the resistance of mean vs. median? Which is more affected by the outlier?

Mean
Median
Both equally
Neither

Solution:

Mean = 25.2, but median = 7. The mean is heavily affected by 100; median is resistant.

Question 12

A survey has \( Q_1 = 20, Q_3 = 60 \). A value of 130 is:

Within the IQR
An outlier (above \( Q_3 + 1.5 \cdot \mathrm{IQR} \))
Within 1.5 IQR of Q3
Below the median

Solution:

IQR = 40. Upper fence = \( 60 + 1.5(40) = 120 \). Since 130 > 120, it is an outlier.