Question

1. candidate a got 260 votes, candidate b 210 votes, candidate c 30 votes in an

exit poll. assuming a representative sample, can you declare a the winner yet?

1. a coin is tossed 100 times and only 37 heads come up. is this coin unfair or could

this just happen by reasonable chance?

1. find the mean, standard deviation, five - number summary, and determine outliers

for the following data: 20, 22, 25, 35, 36, 48, 49, 50, 78, 100.

1. draw a box plot for the data from problem 19.

1. you go out in a boat and capture 30 fish, then release them. the next day you

capture 50 fish and 2 are tagged. based on this one sample, how many fish would
you estimate are in the lake?

1. given test a has an average score of 84 with s = 6 and test b has an average

score of 66 with s = 9, for a score of 92 on test a find its equivalent comparative
score on test b.

Explanation:

Problem 17

Step1: Calculate total votes

Total votes = 260 + 210 + 30 = 500

Step2: Check if A has majority

Majority requires more than 50% of total votes. 50% of 500 is 250. A has 260 votes, which is more than 250.

Step1: Theoretical probability of heads

For a fair coin, P(heads) = 0.5. Expected number of heads in 100 tosses: 100 * 0.5 = 50.

Step2: Analyze observed vs expected

Observed heads = 37, which is much less than 50. We can use binomial probability or normal approximation to check the probability of getting 37 or fewer heads. The z - score for X = 37 is $z=\frac{37 - 50}{\sqrt{100*0.5*0.5}}=\frac{- 13}{5}=-2.6$. The probability of Z < - 2.6 is about 0.0047, which is very low (less than 5% usually considered significant). So the coin is likely unfair.

Step1: Calculate the mean

Mean ($\bar{x}$) = $\frac{20 + 22+25 + 35+36 + 48+49+50+78+100}{10}=\frac{463}{10} = 46.3$

Step2: Sort the data

Sorted data: 20, 22, 25, 35, 36, 48, 49, 50, 78, 100

Step3: Find five - number summary

• Minimum: 20
• Q1 (25th percentile): The median of the first 5 numbers (20, 22, 25, 35, 36). Median of these is 25.
• Median (50th percentile): The average of the 5th and 6th numbers. $\frac{36 + 48}{2}=42$
• Q3 (75th percentile): The median of the last 5 numbers (48, 49, 50, 78, 100). Median of these is 50.
• Maximum: 100

Step4: Calculate standard deviation

First, find the squared differences from the mean:
$(20 - 46.3)^2=(-26.3)^2 = 691.69$
$(22 - 46.3)^2=(-24.3)^2 = 590.49$
$(25 - 46.3)^2=(-21.3)^2 = 453.69$
$(35 - 46.3)^2=(-11.3)^2 = 127.69$
$(36 - 46.3)^2=(-10.3)^2 = 106.09$
$(48 - 46.3)^2=(1.7)^2 = 2.89$
$(49 - 46.3)^2=(2.7)^2 = 7.29$
$(50 - 46.3)^2=(3.7)^2 = 13.69$
$(78 - 46.3)^2=(31.7)^2 = 1004.89$
$(100 - 46.3)^2=(53.7)^2 = 2883.69$

Sum of squared differences: $691.69+590.49 + 453.69+127.69+106.09+2.89+7.29+13.69+1004.89+2883.69 = 5882.1$

Variance ($s^2$)=$\frac{5882.1}{10 - 1}=\frac{5882.1}{9}\approx653.57$

Standard deviation ($s$)=$\sqrt{653.57}\approx25.56$

Step5: Check for outliers

IQR = Q3 - Q1 = 50 - 25 = 25

Lower bound = Q1 - 1.5*IQR = 25 - 37.5=-12.5

Upper bound = Q3 + 1.5*IQR = 50+37.5 = 87.5

The data point 100 is greater than 87.5, so 100 is an outlier. 78 is less than 87.5, so no other outliers.

Answer:

Yes, we can declare A the winner as A has more than 50% (260 > 250) of the total votes (500) in the representative sample.

Problem 18