Question

THANKSGIVING SPECIAL!!! For this applied assignment we will be looking at the number of turkeys and bottles of wine sold each November, comparing the American northeast (i.e., our sample) to the total American averages (i.e., the population).

Every Thanksgiving, an average of 960,000 turkeys and 90,000 bottles of wine are sold per state. Importantly, the population variance/standard deviation is not known. Below is the number of turkeys and bottles of wine sold in each of the ten northeast states.

State	# of Turkeys	# Bottles of Wine
Maine	199,700	45,900
Vermont	93,800	25,500
New Hampshire	200,200	60,000
Massachusetts	1,020,100	294,250
Rhode Island	200,000	74,000
Connecticut	525,100	111,250
New York	2,962,500	780,600
Pennsylvania	1,920,200	480,000
New Jersey	1,341,600	355,400
Delaware	142,500	105,500

1. What are the upper and lower boundaries of the range of μ-values that you can say with 95% confidence contains the µ represented by our sample mean for the number of turkeys sold? For our sample mean of the number of bottles of wine sold?
2. If we are interested in whether or not the number of bottles of wine sold in the American northeast increases as the number of turkeys sold increases, how would we write the null and alternative hypotheses (using the new statistical notation for stating statistical hypotheses; see page 235)?
3. What is the strength and direction of the relationship between the number of turkeys and bottles of wine sold in the American northwest?
4. What is r-crit and r-obt for the hypothesis test stated in question #8? Is the hypothesis test significant or insignificant? Would you reject or fail to reject the null hypothesis assuming an alpha-level of 0.05?
5. If you were to use this relationship (and linear regression) to predict how many bottles of wine will be sold in a northeast state in 2018 given the number of turkeys sold in that state, how much more accurate would you be (in percent) compared to if you did not use the relationship to make this prediction (round to two decimal places)?

Answer 1

In order to solve this question we use R software.

R codes and output:

> d=read.table('data.csv',header=T,sep=',')

> head(d)

          State Turkeys   Wine

1         Maine 199700 45900

2       Vermont   93800 25500

3 New Hampshire 200200 60000

4 Massachusetts 1020100 294250

5 Rhode Island 200000 74000

6   Connecticut 525100 111250

> attach(d)

> t.test(Turkeys)

            One Sample t-test

data: Turkeys

t = 2.8299, df = 9, p-value = 0.01973

alternative hypothesis: true mean is not equal to 0

95 percent confidence interval:

172642.1 1548497.9

sample estimates:

mean of x

   860570

> t.test(Wine)

            One Sample t-test

data: Wine

t = 3.0001, df = 9, p-value = 0.01495

alternative hypothesis: true mean is not equal to 0

95 percent confidence interval:

57372.82 409107.18

sample estimates:

mean of x

   233240

> cor.test(Turkeys,Wine,alternative='greater')

            Pearson's product-moment correlation

data: Turkeys and Wine

t = 26.575, df = 8, p-value = 2.161e-09

alternative hypothesis: true correlation is greater than 0

95 percent confidence interval:

0.9806611 1.0000000

sample estimates:

      cor

0.9943839

> fit=lm(Wine~Turkeys)

> summary(fit)

Call:

lm(formula = Wine ~ Turkeys)

Residuals:

   Min     1Q Median     3Q    Max

-36710 -17709 -2745 11939 54802

Coefficients:

             Estimate Std. Error t value Pr(>|t|)   

(Intercept) 1.447e+04 1.200e+04   1.206    0.262   

Turkeys     2.542e-01 9.566e-03 26.575 4.32e-09 ***

---

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 27600 on 8 degrees of freedom

Multiple R-squared: 0.9888, Adjusted R-squared: 0.9874

F-statistic: 706.2 on 1 and 8 DF, p-value: 4.323e-09

Que.1

95% confidence interval for number of turkeys sold is (172642.1 , 1548497.9).

95% confidence interval for number of wine bottle sold is (0.9755 , 0.9987)

Que.2

Hypothesis:

Que.3

Strength : There is high degree positive correlation between number of turkeys sold and number of wine bottle sold.

Que.4

Correlation coefficient = r = 0.9943839

Test statistic, t = 26.575, df = 8, p-value = 2.161e-09

Critical value = 0.632

Since p-value is less than 0.05, hence we reject null hypothesis and conclude that as number of turkeys sold then number of wine bottle sold are increased.