Jenny, a first-year Nutrition Studies student at Hillside College, is conducting research on various common food items and their nutritional facts. She compiled a data set that contains the nutrition facts on 30 common food items. Jenny feels that many of the nutritional facts of these food items are highly correlated and wonders if there is a way to reduce the dimensionality of the data set by creating principal components. A portion of the data set is shown in the accompanying table. (Values are based on 100 grams of the food items.) 

a. Conduct principal component analysis on all the variables except the Name variable. Should you standardize the data? Explain. 

b. How many principal components are computed? What percent of the total variability is accounted for by the first principal component? How many principal components must be retained in order to account for at least 80% of the total variance in the data? 

c. Display the weights used to compute the first principal component scores. 

d. Create a new data set that contains the names of the food items and the principal components that account for at least 80% of the total variance in the data.

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Name Calories Banana 89 155 Egg ⠀ Tuna 184 Total fat 0.3 11 ⠀ 6 ⠀ Protein 1.1 13 ⠀ 30