1. The variance of a data set is the arithmetic mean of the squared differences between the elements of the data set and the arithmetic mean of the data set. For example the arithmetic mean of the data set consisting of 2 7 and 9 is (2+7+9)/3 which is 6 and the variance is [(2-6)^2 + (7-6)^2 + (9-6)^2]/3 which is 26/3. What is the variance of the data set consisting of 3 4 5 6 and 7?
B. {3/2}
C. {6/5}
D. {5/2} Correct
E. 2
Explanation
The data set has 5 numbers, so we are using population variance (divide by n). First find the mean: (3+4+5+6+7)/5 = 25/5 = 5. Now compute the squared deviations from the mean: (3-5)² = 4, (4-5)² = 1, (5-5)² = 0, (6-5)² = 1, (7-5)² = 4. Sum of squared deviations = 4+1+0+1+4 = 10. Variance = 10/5 = 2.
2. A certain store charges for photocopies by the number of pages. The charge is 14 cents per page for the first 5 pages 8 cents per page for each additional page after the first 5 but before the 11th page and 6 cents per page for each additional page after the first 10 pages. How many cents less is the average charge per page for photocopies of 25 pages than for photocopies of 6 pages?
A. 7
B. 6
C. 5 Correct
D. 4
E. 3
Explanation
For 6 pages: first 5 at 14¢ → 70¢, 6th at 8¢ → total 78¢ → average 78/6 = 13¢ per page. For 25 pages: first 5 at 14¢ = 70¢, next 5 at 8¢ = 40¢, remaining 15 at 6¢ = 90¢ → total 200¢ → average 200/25 = 8¢ per page.Difference: 13 − 8 = 5 cents less
3. Larry began reading a 392-page novel on Sunday and finished it the following Saturday. Each day after the first Larry read 12 pages more than he did the previous day. How many pages did Larry read on Sunday?
A. 20 Correct
B. 22
C. 24
D. 26
E. 28
Explanation
Let Sunday = s pages. The seven days form an arithmetic sequence: s, s+12, s+24, s+36, s+48, s+60, s+72. Sum = 7s + (0+12+24+36+48+60+72) = 7s + 252 = 392. So 7s = 140 → s = 20. Larry read 20 pages on Sunday.
4. A manufacturer regularly receives shipments of computer chips. Over the past year the mean number of computer chips per shipment was 1200. If a shipment of 1456 chips was 1.6 standard deviations above the mean how many standard deviations below the mean was a shipment of 848 chips?
A. 1.3
B. 1.4
C. 1.7
D. 2
E. 2.2 Correct
Explanation
Mean = 1,200 chips. A shipment of 1,456 is 1.6 standard deviations above the mean: 1,456 = 1,200 + 1.6σ → 256 = 1.6σ → σ = 256/1.6 = 160.Now 848 chips: 1,200 − 848 = 352 → 352/160 = 2.2 standard deviations below the mean.
5. A certain tour company bought canoes kayaks and life jackets for a total of $3279. If the company paid $65 for each canoe $40 for each kayak and $2 for each life jacket then the company must have bought which of the following? I. An odd number of canoes II. An even number of kayaks III. An even number of life jackets
A. I only Correct
B. II only
C. III only
D. I and III only
E. I II and III
Explanation
65c + 40k + 2l = 3,279.65c is always odd (65×any integer), 40k and 2l are always even → total is odd only if c is odd → statement I must be true. Statements II and III are possible but not necessary.