Question

sorted data set: 3 5 5 6 7 7 7 7 8 8 9 9 10
#24 what is the first quartile q₁ for this data set?
(a) 5.5 (b) 6.5 (c) 7.0 (d) 8.5 (e) none of these
#25 what is the second quartile q₂ for this data set?
(a) 5.5 (b) 6.5 (c) 7.0 (d) 8.5 (e) none of these
#26 what is the third quartile q₃ for this data set?
(a) 5.5 (b) 6.5 (c) 7.0 (d) 8.5 (e) none of these
#27 what is the inter - quartile range iqr for this data set?
(a) 2.0 (b) 2.5 (c) 3.0 (d) 3.5 (e) none of these
#28 what is the variance of this data set? note: please round your answer to 3 digits.
(a) 3.250 (b) 3.333 (c) 3.667 (d) 3.750 (e) none of these
#29 what is the standard deviation of this data set?
(a) 1.803 (b) 1.826 (c) 1.915 (d) 1.936 (e) none of these

Explanation:

Step1: Count data points

Number of data points $n=14$.

Step2: Find Q1 position

Position of $Q_1 = \frac{1}{4}(n+1) = \frac{15}{4}=3.75$. Value: $5 + 0.75*(6-5)=5.75$ → E. None of these? Wait no, wait sorted data: 3,5,5,6,7,7,7,7,8,8,9,9,10? Wait wait, count again: 3,5,5,6,7,7,7,7,8,8,9,9,10 is 13 points. Oh, 13 points: $n=13$.

Step2 corrected: Q1 position for n=13

$Q_1$ position: $\frac{1}{4}(13+1)=3.5$. Value: $\frac{5+6}{2}=5.5$ → A. 5.5.

Step3: Q2 position

$Q_2$ position: $\frac{2}{4}(13+1)=7$. Value: 7 → C.7.0.

Step4: Q3 position

$Q_3$ position: $\frac{3}{4}(13+1)=10.5$. Value: $\frac{8+9}{2}=8.5$ → D.8.5.

Step5: IQR calculation

$IQR=Q3-Q1=8.5-5.5=3.0$ → C.3.0.

Step6: Mean calculation

Mean $\mu=\frac{3+5+5+6+7+7+7+7+8+8+9+9+10}{13}=\frac{91}{13}=7$.

Step7: Variance calculation

Variance $\sigma^2=\frac{\sum(x_i-\mu)^2}{13}$: $(3-7)^2=16, 2*(5-7)^2=8, (6-7)^2=1,4*(7-7)^2=0,2*(8-7)^2=2,2*(9-7)^2=8,(10-7)^2=9$. Sum=16+8+1+0+2+8+9=44. Variance=44/13≈3.3846? Wait no, wait 13 points: 3,5,5,6,7,7,7,7,8,8,9,9,10: sum of squares: 9+25+25+36+49+49+49+49+64+64+81+81+100= 9+252=59, +36=95, +494=196→291, +642=128→419, +812=162→581, +100=681. Variance (population): (681/13)-7²=52.3846-49=3.3846≈3.385, sample variance: 44/12≈3.6667→C.3.667.

Step8: Standard deviation

Sample std dev: sqrt(3.6667)≈1.915→C.1.915.

Wait, so answers: #24:A, #25:C, #26:D, #27:C, #28:C, #29:C.

Wait #28: option C is 3.667 which is sample variance (44/12≈3.6667). So yes, C.

Final Answer:

#24: A. 5.5
#25: C. 7.0
#26: D. 8.5
#27: C. 3.0
#28: C. 3.667
#29: C. 1.915

Step1: Count data points

n=13 (sorted:3,5,5,6,7,7,7,7,8,8,9,9,10).

Step2: Q1 calculation

Position=3.5 → (5+6)/2=5.5.

Step3: Q2 calculation

Position=7 → value=7.

Step4: Q3 calculation

Position=10.5 → (8+9)/2=8.5.

Step5: IQR calculation

8.5-5.5=3.0.

Step6: Sample variance

Sum of squared deviations=44 → 44/12≈3.667.

Step7: Sample std dev

sqrt(3.667)≈1.915.

So the answers are as above.

Answer:

A. 5.5, C. 7.0, D. 8.5, C. 3.0, C. 3.667, B. 1.826