Question

Many consumers pay careful attention to stated nutritional contents on packaged foods when making purchases. It is therefore important that the information on packages be accurate. A random sample of

n = 12 frozen dinners of a certain type was selected from production during a particular period, and the calorie content of each one was determined. (This determination entails destroying the product, so a census would certainly not be desirable!) Here are the resulting data:

255

244

239

242

265

245

259

248

225

226

251

233

[a] Use R (or Excel) to find the summary statistics of the data.                                                         [1 pt]

Copy and paste your computer printout here (or to a blank MS Word document).

[b] Construct a boxplot for the data and comment on any interesting features.                                [4 pt]

Copy and paste your computer printout here (or to a blank MS Word document).

[c] Construct a normal probability plot (QQplot as well as qqline) and comment on any interesting features.                                                                                                                                            [4 pt]

Copy and paste your computer printout here (or to a blank MS Word document).

[d] Use an appropriate test to ensure the data set satisfies the normality assumption.                      [3 pt]

What is your conclusion regarding the data set?

[e] Is it reasonable to test hypotheses about mean calorie content m by using a t-test? Explain why or why not.                                                                                                                                       

Answer:           Yes                  No                   E Circle one                                                              [1 pt]

Explanation:                                                                                                                                       [3 pt]

[e] The stated calorie content is m = 240. Does the boxplot in [a] suggest the true mean content differs from the stated value? Explain your reasoning.                                                                               [3 pt]

[f] Carry out a formal test of the hypotheses suggested in part [e]                                                   

Write your hypotheses in symbols and in words.                                                                               [8 pt]

Use statistical software (such as R) to get analysis results. State what the results tell you about the data.                                                                                                                                                               [8 pt]

Answer 1

Solution:

summary statistics are:

length(caloriecontent)
12
mean(caloriecontent)
244.3333
sd(caloriecontent)
12.38278
summary(caloriecontent)
Min. 1st Qu. Median Mean 3rd Qu. Max.
225.0 237.5 244.5 244.3 252.0 265.0

From sumamry

mean=244.333

standard deviation=12.38278

sample szie=n=12

Min=225

Q1=237.5

median=244.5

Q3=252.0

max=265.0

Solutionb:

Rcode:

caloriecontent <- c(255,
  
244,
  
239,
  
242,
  
265,
  
245,
  
259,
  
248,
  
225,
  
226,
  
251,
  
233)
boxplot(caloriecontent,main="boxplot for frozen dinners -caloriecontent")
abline(h = min(caloriecontent), col = "Blue")
abline(h = max(caloriecontent), col = "Yellow")
abline(h = median(caloriecontent), col = "Green")
abline(h = quantile(caloriecontent, c(0.25, 0.75)), col = "Red")

fivenum(caloriecontent)

From bxoplot we observe

no outilers

Fiveumber sumamry is

225.0 236.0 244.5 253.0 265.0

MIN=225

Q1=236

Q2=244.5

Q3=253

MAX=265

Solutionc:

Rcode:

qqnorm(caloriecontent)
qqline(caloriecontent)

From normal QQ plot we observe

sample follows normal distribution

as most of the points fall on the straight line

[d] Use an appropriate test to ensure the data set satisfies the normality assumption.   

perform shapiro .test

Null hypothesis:Data comes from normal distribution

Alternative hypothesis:Data do not come from normal distribution

Rcode:

shapiro.test(caloriecontent)

output:

   Shapiro-Wilk normality test

data: caloriecontent
W = 0.97361, p-value = 0.9447

test statistic=0.97361

p=0.9447

p>0.05

Fail to reject Ho.

Accept H0.
There is suffcient evidence to conclude that the data comes from normal distribution.