2. Suppose we want to determine the (binomial) probability(p) of getting 6 heads in 15 flips of a 2-sided coin. Using the Binomial Probabilities Table in Appendix B of the text, what values of n, x and p would we use to look up this probability, and what would be the probability?

n = 15, x = 6, p = 0.5
Probability = 15C6 0.5^6 0.5^9 = 0.1527

3. For the table that follows, answer the following questions: x y 1 -1 2 -5 3 -9 4 - Would the correlation between x and y in the table above be positive or negative? - Find the missing value of y in the table. - How would the values of this table be interpreted in terms of linear regression? - If a “line of best fit” is placed among these points plotted on a coordinate system, would the slope of this line be positive or negative?

Negative (because as x increases y decreases)
y = -13(because we see that as x increases by 1, y decreases by 4)
The values in the table when plotted will give straight line graph. The fit will be perfectly linear.
The slope would be negative

4. Answer the following: - If the correlation coefficient is 0.02, what is the sign of the slope of the regression line?

Positive (since r is 0.02 which is positive - Sign of the slope is the same as that of the correlation coefficient)

5. As the correlation coefficient increases from 0.30 to 0.34, do the points of the scatter plot move toward the regression line, or away from it?

Towards the line (since there is much stronger association between x and y now)

6. Determine whether each of the distributions given below represents a probability distribution. Justify your answer. x 1 2 3 4 P(x) 1/4 5/12 1/3 1/6 (A) x 3 6 8 P(x) 0.1 3/5 0.3 (B) x 20 30 40 50 P(x) 0.2 -0.2 0.7 0.3 (C)

Here, we use the fact that for a probability distribution to be valid, the sum of P(x) should be 1. Also no P(x) canlie outside the limits 0 and 1.
(a) No, since the sum of all P(x) values is more than 1
(b) Yes
(c) No, since probability can’t be negative

7. Three cards are selected, one at a time from a standard deck of 52 cards. Let x represent the number of eights drawn in a set of 3 cards. (A) If this experiment is completed without replacement, explain why x is not a binomial random variable. (B) If this experiment is completed with replacement, explain why x is a binomial random variable.

If 3 cards are selected one at a time, then for the first draw, the probability of drawing an eight is 4/52 = 1/13. But for subsequent draws the probability of drawing an eight does not remain as 4/52 = 1/13 since, for example, for the second draw, the probability of drawing an eight will be 4/51 (If the first draw did not give an Ace) or 3/51 = 1/17 (If the first draw gave an eight). Thus, the probability of drawing an eight is not the same in every trial. Therefore, x is not a binomial variable.
(b) In this case, the probability of drawing an eight remains the same (4/52 = 1/13) in every trial. Therefore x is a binomial variable.

8. You are given the following data. Number of Absences Final Grade 0 93 1 92 2 79 3 68 4 64 5 56 - Find the correlation coefficient for the data. - Find the equation for the regression line for the data, and predict the final grade of a student who misses 3.5 days.

Number of absences (x)Final Grade (y)
093
192
279
368
464
556
We use excel to get the scatter plot, correlation coefficient and regression equation.

(a)
The points are scattered about a straight line. It appears from the scatter plot that the data are linearly related.
The correlation coefficient is r = -0.9829
There is a strong negative correlation between x and y. As x increases, y decreases.
(b) The equation is y = -8x + 95.333. The line is shown on the scatter plot.
(c) When x = 3.5, y = -8(3.5) + 95.333 = 67.333