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I need help with some statistical questions.

1. A regression equation is given by [tex]Y = 20 + 0.75x[/tex]. What is the value of the residual for the observed data point [tex]x = 100[/tex] and [tex]y = 90[/tex]?

2. Data obtained from a number of women's clothing stores show that there is a linear relationship between sales ([tex]y[/tex], in dollars) and advertising budget ([tex]x[/tex], in dollars). The regression equation was found to be [tex]y = 5000 + 7.50x[/tex]. If the advertising budgets of two women's clothing stores differ by $30,000, what will be the predicted difference in their sales?

4. A regression analysis between sales ([tex]y[/tex], in $1000) and price ([tex]x[/tex], in dollars) resulted in the following equation: [tex]y = 50,000 - Bx[/tex]. The above equation implies that an increase of [tex]___\$?___[/tex] in price is associated with a decrease of [tex]___\$?___[/tex] in sales. (Fill in the blanks in dollars.)

5. Suppose the correlation coefficient between height (measured in feet) and weight (measured in pounds) is 0.40. What is the correlation coefficient between height measured in inches and weight measured in ounces? (Note: one foot = 12 inches, one pound = 16 ounces.)

Answer :

1. The equation is Y = 20 + 0.75x For the given values, x = 100 and y = 90 Therefore, the fitted value linear equation

Y = 20 + 0.75*100 = 95

Residual value = Observed value - Fitted value = 90 - 95 = -5

Therefore, the residual value is -5.

2. Given that sales (y) and advertising budget (x) are related by the equation, y = 5000 + 7.5x.
If the advertising budgets of two women's clothing stores differ by $30,000, then the difference in their predicted sales can be found as follows:
Let the advertising budgets of the two stores be x1 and x2.
Then the predicted sales for the two stores will be y1 = 5000 + 7.5x1 and y2 = 5000 + 7.5x2.
The difference in their predicted sales will be:
y2 - y1 = (5000 + 7.5x2) - (5000 + 7.5x1) = 7.5(x2 - x1)
Since the difference in their advertising budgets is $30,000, we have:
x2 - x1 = 30,000
Therefore, the predicted difference in their sales is 7.5(30,000) = $225,000.

3. An increase of $1 in price is associated with a decrease of $B in sales.
Here, the regression equation is y = 50,000 - Bx.
Since the coefficient of x is negative, we can conclude that the relationship between sales and price is negative or inverse.
Therefore, if the price increases, the sales will decrease.
The coefficient B gives the rate at which sales decrease for a unit increase in price.
Here, the coefficient B is not given in the question.

4. Let the correlation coefficient between height and weight be r1. We have the formula for the correlation coefficient as follows:
r = Covariance(X, Y) / (StdDev(X) * StdDev(Y))
We are given that the correlation coefficient between height and weight is r1 = 0.40.
We need to find the correlation coefficient between height measured in inches and weight measured in ounces.
Let h1 and w1 be the height (in inches) and weight (in ounces) of the first person.
Then we have h2 = 12h1 and w2 = 16w1 for the same person measured in feet and pounds.
Therefore, we have:
Covariance(h1, w1) = Covariance(12h1, 16w1) = 12 * 16 Covariance(h1, w1) = 192 Covariance(h1, w1)
StdDev(h1) = StdDev(12h1) = 12 StdDev(h1)
StdDev(w1) = StdDev(16w1) = 16 StdDev(w1)
Substituting these values in the formula for correlation coefficient, we get:
r2 = Covariance(h1, w1) / (StdDev(h1) * StdDev(w1)) = r1 * 192 / (12 * 16) = 0.40 * 12 / 16 = 0.30
Therefore, the correlation coefficient between height measured in inches and weight measured in ounces is 0.30.

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