Hypothesis Testing

Lab4 - Unit 3

Email             : sausan.ramadhani@student.matanauniversity.ac.id
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Jurusan          : Statistika
Address         : ARA Center, Matana University Tower
                         Jl. CBD Barat Kav, RT.1, Curug Sangereng, Kelapa Dua, Tangerang, Banten 15810.

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Exercise 1

Right Tail: A food company argue that for each a cookie bag of their products, there is at most 2 grams of saturated fat in a single cookie. In a sample of 40 cookies, it is found that the mean amount of saturated fat per cookie is 2.1 grams. Assume that the population standard deviation is 0.25 grams. At .05 significance level, can we reject the claim?

Answer

mu0 = 2
n = 40
xbar = 2.1
sigma = 0.25
z = (xbar-mu0)/(sigma/sqrt(n))
z

## [1] 2.529822

alpha = 0.05
z.alpha = qnorm(1-alpha)
z.alpha

## [1] 1.644854

pnorm(z, lower.tail = FALSE)      # Z test in right tailed

## [1] 0.005706018

library(BSDA)
zsum.test(mean.x=xbar, sigma.x = sigma, n.x = n,  
          alternative = "greater", mu = mu0,
          conf.level = 0.95)

## 
##  One-sample z-Test
## 
## data:  Summarized x
## z = 2.5298, p-value = 0.005706
## alternative hypothesis: true mean is greater than 2
## 95 percent confidence interval:
##  2.034981       NA
## sample estimates:
## mean of x 
##       2.1

Exercise 2

To test the hypothesis that the mean systolic blood pressure in a certain population equals 140 mmHg. The standard deviation has a known value of 20 and a data set of 55 patients is available.

#Blood_pressure dataset
no <- seq(1:55)
status <- c(rep(0, 25), rep(1, 30))
mmhg <- c(120,115,94,118,111,102,102,131,104,107,115,139,115,113,114,105,
          115,134,109,109,93,118,109,106,125,150,142,119,127,141,149,144,
          142,149,161,143,140,148,149,141,146,159,152,135,134,161,130,125,
          141,148,153,145,137,147,169)
blood_pressure <-data.frame(no,status,mmhg)
blood_pressure

##    no status mmhg
## 1   1      0  120
## 2   2      0  115
## 3   3      0   94
## 4   4      0  118
## 5   5      0  111
## 6   6      0  102
## 7   7      0  102
## 8   8      0  131
## 9   9      0  104
## 10 10      0  107
## 11 11      0  115
## 12 12      0  139
## 13 13      0  115
## 14 14      0  113
## 15 15      0  114
## 16 16      0  105
## 17 17      0  115
## 18 18      0  134
## 19 19      0  109
## 20 20      0  109
## 21 21      0   93
## 22 22      0  118
## 23 23      0  109
## 24 24      0  106
## 25 25      0  125
## 26 26      1  150
## 27 27      1  142
## 28 28      1  119
## 29 29      1  127
## 30 30      1  141
## 31 31      1  149
## 32 32      1  144
## 33 33      1  142
## 34 34      1  149
## 35 35      1  161
## 36 36      1  143
## 37 37      1  140
## 38 38      1  148
## 39 39      1  149
## 40 40      1  141
## 41 41      1  146
## 42 42      1  159
## 43 43      1  152
## 44 44      1  135
## 45 45      1  134
## 46 46      1  161
## 47 47      1  130
## 48 48      1  125
## 49 49      1  141
## 50 50      1  148
## 51 51      1  153
## 52 52      1  145
## 53 53      1  137
## 54 54      1  147
## 55 55      1  169

Answer

mu0 = 140
n = 55
xbar = mean(blood_pressure$mmhg)
sigma = 20
z = (xbar-mu0)/(sigma/sqrt(n))
z

## [1] -3.708099

alpha = .05                                            # .05 significance level
z.half.alpha = qnorm(1-alpha/2)                        # per-one tail .025 significance level
c(-z.half.alpha, z.half.alpha)

## [1] -1.959964  1.959964

2*pnorm(-abs(z))          # Z test in two tails

## [1] 0.0002088208

zsum.test(mean.x=xbar, sigma.x = 20, n.x = 55,  
          alternative = "two.sided", mu = mu0,
          conf.level = 0.95)

## 
##  One-sample z-Test
## 
## data:  Summarized x
## z = -3.7081, p-value = 0.0002088
## alternative hypothesis: true mean is not equal to 140
## 95 percent confidence interval:
##  124.7144 135.2856
## sample estimates:
## mean of x 
##       130

Exercise 3

Garuda-food Indonesia claims that for each a cookie bag states of their product, there is at most 2 grams of saturated fat in a single cookie. In a sample of 40 cookies, it is found that the mean amount of saturated fat per cookie is 2.1 grams. Assume that the sample standard deviation is 0.3 gram. At .05 significance level, can we reject the claim?

Answer

mu0 = 2
n = 40
xbar = 2.1
s = 0.3
t = (xbar-mu0)/(s/sqrt(n))
t

## [1] 2.108185

alpha = 0.05
t.alpha = qt(1-alpha, df = n-1)
t.alpha

## [1] 1.684875

pt(t, df=n-1, lower.tail = FALSE)         # t test in right tailed

## [1] 0.020746

Exercise 4

To test the hypothesis that the mean systolic blood pressure in a certain population equals 140 mmHg. The standard deviation has a known value of 20 and a data set of 55 patients is available.

#Blood_pressure dataset
no <- seq(1:55)
status <- c(rep(0, 25), rep(1, 30))
mmhg <- c(120,115,94,118,111,102,102,131,104,107,115,139,115,113,114,105,
          115,134,109,109,93,118,109,106,125,150,142,119,127,141,149,144,
          142,149,161,143,140,148,149,141,146,159,152,135,134,161,130,125,
          141,148,153,145,137,147,169)
blood_pressure <-data.frame(no,status,mmhg)
blood_pressure

##    no status mmhg
## 1   1      0  120
## 2   2      0  115
## 3   3      0   94
## 4   4      0  118
## 5   5      0  111
## 6   6      0  102
## 7   7      0  102
## 8   8      0  131
## 9   9      0  104
## 10 10      0  107
## 11 11      0  115
## 12 12      0  139
## 13 13      0  115
## 14 14      0  113
## 15 15      0  114
## 16 16      0  105
## 17 17      0  115
## 18 18      0  134
## 19 19      0  109
## 20 20      0  109
## 21 21      0   93
## 22 22      0  118
## 23 23      0  109
## 24 24      0  106
## 25 25      0  125
## 26 26      1  150
## 27 27      1  142
## 28 28      1  119
## 29 29      1  127
## 30 30      1  141
## 31 31      1  149
## 32 32      1  144
## 33 33      1  142
## 34 34      1  149
## 35 35      1  161
## 36 36      1  143
## 37 37      1  140
## 38 38      1  148
## 39 39      1  149
## 40 40      1  141
## 41 41      1  146
## 42 42      1  159
## 43 43      1  152
## 44 44      1  135
## 45 45      1  134
## 46 46      1  161
## 47 47      1  130
## 48 48      1  125
## 49 49      1  141
## 50 50      1  148
## 51 51      1  153
## 52 52      1  145
## 53 53      1  137
## 54 54      1  147
## 55 55      1  169

Answer

mu0 = 140
n = 55
xbar = mean(blood_pressure$mmhg)
s = 20
t = (xbar-mu0)/(s/sqrt(n))
t

## [1] -3.708099

alpha = .05                                            # .05 significance level
z.half.alpha = qnorm(1-alpha/2)                        # per-one tail .025 significance level
c(-z.half.alpha, z.half.alpha)

## [1] -1.959964  1.959964

2*pt(t, df=n-1)

## [1] 0.0004938793

Exercise 5

Right tail: Garuda-food Indonesia claims that for each a cookie bag states of their product, there is at most 2 grams of saturated fat in a single cookie. Assume the actual mean amount of saturated fat per cookie is 2.075 grams and the sample standard deviation is 0.25 grams. At .05 significance level, what is the probability of having a type II error for a sample size of 35 cookies?

Answer

mu0 = 2
mu = 2.075
sigma = 0.25
alpha = 0.05
n = 35

sem = sigma/sqrt(n)
sem

## [1] 0.04225771

q = qnorm(alpha, mean=mu0, sd=sem)
q

## [1] 1.930492

pnorm(q, mean=mu, sd=sem, lower.tail = FALSE)

## [1] 0.9996865

Exercise 6

Under same assumptions as case 27, if actual mean population weight is 14.9 kg, what is the probability of type II errors? What is the power of the hypothesis test?

Answer

mu0 = 15.4
mu = 14.9
sigma = 2.5
alpha = 0.05
n = 35

sem = sigma/sqrt(n)
sem

## [1] 0.4225771

I = c(alpha/2, 1-alpha/2)
q = qnorm(I, mean=mu0, sd=sem)
q

## [1] 14.57176 16.22824

p = pnorm(q, mean=mu, sd=sem)
p

## [1] 0.2186537 0.9991644