chapter 2 | number sense: basic computational skills and benchmarks

f. suppose we create a new data set that we will call data set 2. in this data set we have eighteen persons—eight with a wingspan of 80.5 inches, five with a wingspan of 75.5, and five with a wingspan of 85.5. what is the mean of data set 2?

i. 75.5
ii. 80.5
iii. 85.5
iv. 90.5

g. suppose in a data set we call data set 2, we have eighteen persons—eight all with a wingspan of 80.5 inches, five all with a wingspan of 75.5, and five all with a wingspan of 85.5. what is the median of data set 2?

i. 75.5
ii. 80.5
iii. 85.5
iv. 90.5
histogram: wingspan data set 2, with bars labeled a, b, c, d on x-axis (wingspan inches), y-axis frequency
h. we have organized the wingspans from data set 1 into the histogram you see here. on the histogram, we have the letters a, b, c, and d superimposed on the horizontal axis. which letter identifies the interval containing the location of the mean of the 18 wingspans in data set 2?

remember how to read the axis. the bar above the 80, for example, is the count in the interval from 75 to 80.

i. a
ii. b
iii. c
iv. d
i. how adequate would the mean value alone be for distinguishing the data in data set 1 from those in data set 2? explain.
j. compare the histograms from data sets 1 and 2 (both shown above). specify at least two ways that the histograms are notably different with respect to how their data are spread out.
k. recall data set 2, defined above. this data set has eighteen persons—eight all with a wingspan of 80.5 inches, five all with a wingspan of 75.5, and five all with a wingspan of 85.5. you get to pick two wingspans at random from this data set, eyes closed. call your choices ( x_1 ) and ( x_2 ). you will receive a payout of ( $(80.5 - x_1)^2 + (80.5 - x_2)^2 ) based on these choices. if it costs you $25.00 to play, what is the maximum profit you can make?

Question

chapter 2 | number sense: basic computational skills and benchmarks

f. suppose we create a new data set that we will call data set 2. in this data set we have eighteen persons—eight with a wingspan of 80.5 inches, five with a wingspan of 75.5, and five with a wingspan of 85.5. what is the mean of data set 2?

i. 75.5
ii. 80.5
iii. 85.5
iv. 90.5

g. suppose in a data set we call data set 2, we have eighteen persons—eight all with a wingspan of 80.5 inches, five all with a wingspan of 75.5, and five all with a wingspan of 85.5. what is the median of data set 2?

i. 75.5
ii. 80.5
iii. 85.5
iv. 90.5
histogram: wingspan data set 2, with bars labeled a, b, c, d on x-axis (wingspan inches), y-axis frequency
h. we have organized the wingspans from data set 1 into the histogram you see here. on the histogram, we have the letters a, b, c, and d superimposed on the horizontal axis. which letter identifies the interval containing the location of the mean of the 18 wingspans in data set 2?

remember how to read the axis. the bar above the 80, for example, is the count in the interval from 75 to 80.

i. a
ii. b
iii. c
iv. d
i. how adequate would the mean value alone be for distinguishing the data in data set 1 from those in data set 2? explain.
j. compare the histograms from data sets 1 and 2 (both shown above). specify at least two ways that the histograms are notably different with respect to how their data are spread out.
k. recall data set 2, defined above. this data set has eighteen persons—eight all with a wingspan of 80.5 inches, five all with a wingspan of 75.5, and five all with a wingspan of 85.5. you get to pick two wingspans at random from this data set, eyes closed. call your choices ( x_1 ) and ( x_2 ). you will receive a payout of ( $(80.5 - x_1)^2 + (80.5 - x_2)^2 ) based on these choices. if it costs you $25.00 to play, what is the maximum profit you can make?

Accepted Answer

### f.
# Explanation:
## Step1: Calculate total sum
Sum = (8×80.5) + (5×75.5) + (5×85.5) = 644 + 377.5 + 427.5 = 1449
## Step2: Calculate mean
Mean = $\frac{1449}{18}$ = 80.5
# Answer: ii. 80.5

### g.
# Explanation:
## Step1: Order data & find middle
18 data points, median is average of 9th and 10th. First 5:75.5, next 8:80.5. 9th and 10th are 80.5.
## Step2: Determine median
Median = 80.5
# Answer: ii. 80.5

### h.
# Explanation:
## Step1: Recall mean from f
Mean is 80.5.
## Step2: Match to interval
From histogram info, B's interval (around 80) contains 80.5.
# Answer: ii. B

### i.
# Brief Explanations:
If Data Set 1 has a different mean, mean alone can distinguish. If means are equal, mean alone is inadequate (needs spread/shape info). Need to know Data Set 1’s mean, but generally, mean alone may not suffice if distributions differ in spread/shape even with same mean.
# Answer: Depends on Data Set 1’s mean. If means differ, adequate; if same, inadequate (e.g., if Data Set 1 has different spread/shape, mean alone can’t distinguish).

### j.
# Brief Explanations:
1. Spread: Data Set 2 has values at 75.5, 80.5, 85.5 (narrower?); Data Set 1’s histogram (from A - D) may have wider intervals. 2. Frequency: Data Set 2 has 8 at 80.5, 5 each at 75.5/85.5; Data Set 1’s histogram bars (A, B, C, D) have different frequencies (e.g., B tallest, A and C shorter, D zero?—from graph).
# Answer: 1. Data Set 2 has values at 75.5, 80.5, 85.5 (less spread in values); Data Set 1’s histogram has intervals (e.g., A:75 - 80, B:80 - 85, etc.) with different spreads. 2. Data Set 2 has 8 at 80.5, 5 each at 75.5/85.5; Data Set 1’s histogram (e.g., B bar tallest, A and C shorter, D empty).

### k.
# Explanation:
## Step1: Find max payout
To maximize $(80.5 - x_1)^2 + (80.5 - x_2)^2$, pick $x_1, x_2$ farthest from 80.5 (75.5 or 85.5). Let $x_1 = x_2 = 75.5$ (or 85.5, same result).
$(80.5 - 75.5)^2 + (80.5 - 75.5)^2 = 5^2 + 5^2 = 25 + 25 = 50$.
Payout = $5×50 = 250$.
## Step2: Calculate profit
Profit = Payout - Cost = 250 - 25 = 225.
# Answer: $225$

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QUESTION IMAGE

Question

Explanation:

f.

Step1: Calculate total sum

Step2: Calculate mean

Step1: Order data & find middle

Step2: Determine median

Step1: Recall mean from f

Step2: Match to interval

Answer:

g.