Question

The table below lists weights​ (carats) and prices​ (dollars) of randomly selected diamonds. Find the​ (a) explained​ variation, (b) unexplained​ variation, and​ (c) indicated prediction interval. There is sufficient evidence to support a claim of a linear​ correlation, so it is reasonable to use the regression equation when making predictions. For the prediction​ interval, use a​ 95% confidence level with a diamond that weighs 0.8 carats.

Weight   Price
0.3   520
0.4   1171
0.5   1336
0.5   1425
1.0   5671
0.7   2278

Answer 1

> weight = c(0.3,.4,.4,.5,1,.7) > price = c(520,1171,1336,1425,5671,2278) > cor(weight,price) [1] 0.9656554 # we see that there is high positive correlation between weight and price > mod = lm(price~weight) > summary(mod) Call: lm(formula = price ~ weight) Residuals: 1 2 3 4 5 6 181.8 141.3 306.3 -296.1 492.7 -826.0 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) -1736.1 556.8 -3.118 0.03560 * weight 6914.5 930.2 7.433 0.00175 ** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 538.4 on 4 degrees of freedom Multiple R-squared: 0.9325, Adjusted R-squared: 0.9156 F-statistic: 55.25 on 1 and 4 DF, p-value: 0.001749 Since R-squared: 0.9325, so the explained variation is 93.25% and unexplained variation is 1- 0.9325 = 0.0675, i,e 6.7% > df1 = data.frame(weight = 0.8) > predict(mod,df1, interval = "confidence") fit lwr upr 1 3795.453 2907.004 4683.902

> The predicted value of price for weight = 0.8 is 3795.453 and the 95% confidence interval is (2907.004, 4683.902)