 Statistics Questions

Subject Mathematics Advanced Statistics

Question

Problem 1.
Suppose we have randomly sampled eight chocolate bars from a chocolatier, which are supposed to contain 90% cocoa on average. We assume that the actual cocoa content follows a normal distribution. Measurement of the actual cocoa content (%) of the sampled bars showed the following results: 94.71 89.86 78.63 84.74 93.71 82.31 80.38 84.01
Solve the following problems using R. Provide (i) your answer to each question, (ii) the R commands that were used, and (iii) the output you got from R.
(a) Is there a reason to believe that the actual cocoa content might be less than 90% based on the above data? Make a decision at a significance level α = 0.1. Justify your answer.
(b) Do we have a good reason to believe that the actual cocoa content might be different from what the chocolatier claims? Make a decision at a significance level α = 0.1. Justify your answer.

Problem 3.
A new library has been built in a humid region. After a year of operation, the library replaced their dehumidifiers. In order to find out whether there was any significant change in average humidity level (%), they measured the humidity level at several locations before and after the installation of new dehumidifiers. Solve the following problems using R. Provide (i) your answer to each question, (ii) the R commands that were used, and (iii) the output you got from R.
(a) Suppose the following results have been obtained by measuring the humidity level at 7 random locations before replacing the dehumidifiers, and then measuring the humidity at another 7 randomly picked locations after the replacement.
before: 32.9 42.3 25.3 36.7 35.5 31.6 29.7
after : 23.3 26.8 39.4 29.7 25.1 21.6 23.2
At a significance level α = 5%, is there a reason to believe that the humidity level may have decreased after replacing the dehumidifiers? Justify your answer.
(b) Suppose we randomly picked 7 locations and then tested the humidity level at each of these locations before and after replacing the dehumidifiers:
location: LOC1 LOC2 LOC3 LOC4 LOC5 LOC6 LOC7
before : 24.6 36.3 25.6 19.7 27.6 27.1 26.0
after : 33.7 34.0 31.3 35.7 23.5 29.0 35.7
At a significance level α = 5%, is there a reason to believe that the humidity level may have increased after the replacement?

Solution Preview

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# Problem 1

x <- c(94.71, 89.86,78.63,84.74,93.71,82.31,80.38,84.01)
# (a)
t.test(x,alternative=c("less"),mu=90,conf.level=0.90)
# One Sample t-test
#
# data: x
# t = -1.8531, df = 7, p-value = 0.05314
# alternative hypothesis: true mean is less than 90
# 90 percent confidence interval:
#   -Inf 89.06459
# sample estimates:
#   mean of x
# 86.04375

# Because the p-value (0.053) is not greater than alpha=0.1, the test
# rejected the hypothesis of mean percentage being less than 90%....

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