Binomial based Tests

The binomial Test -p- Example from Lecture Notes

Example 1:

• From studies 50% of men suffer from side effects of a surgery.
• New method of performing the surgery is tested.
• Out of 19 operations only 3 men suffered from the side effects.

Question: is it safe to conclude that the new method is effective?

To answer the question we run a lower tailed binomial test.

binom.test(x=3, # #successes
          n=19, # total number of operations
          p= 0.5, # hypothesized probability
          alternative="l" # lower tailed test
          )

    Exact binomial test

data:  3 and 19
number of successes = 3, number of trials = 19, p-value = 0.002213
alternative hypothesis: true probability of success is less than 0.5
95 percent confidence interval:
 0.0000000 0.3594256
sample estimates:
probability of success 
             0.1578947 

Example 2:

To answer the Mendelian example from the notes we run a two tailed binomial test.

binom.test(x=682, # #tall plants
          n=925, # total number of plants
          p= 0.75, # hypothesized probability
          alternative="t" # two tailed test
          )

    Exact binomial test

data:  682 and 925
number of successes = 682, number of trials = 925, p-value = 0.3825
alternative hypothesis: true probability of success is not equal to 0.75
95 percent confidence interval:
 0.7076683 0.7654066
sample estimates:
probability of success 
             0.7372973 

The quantile Test -x_p- Example from Lecture Notes

Example 3:

Assume X is the score of a high school examination to go to college. Previous studies showed that the upper quartile of these scores is $X_{0.75} = $ 193.

15 graduate high school students scores are given:

189, 233, 195, 160, 212, 176, 231, 185, 199, 213, 202, 193, 174, 166, 248.

To test whether the population upper quartile is still 193, we run a two-tailed quantile test.

# data
x= c(189, 233, 195, 160, 212, 176, 231, 185, 199, 213, 202, 193, 174, 166, 248)

# Test statistics
T1 = sum(x<=193)
T2 = sum(x<193)

# p-value
p = 2*min(pbinom(T1,15,0.75),pbinom(T2-1,15,0.75,lower.tail = F))
p

[1] 0.03459968

The sign Test - Example from Lecture Notes

Example 4:

• Item A (old process) and Item B (new process)
• Out of 10 consumers:
  • 8 preferred B over A ==> +
  • 1 preferred A over B ==> -
  • 1 had no preference ==> 0
• Do consumers prefer B over A?
• To answer this we run an upper tailed sign test.

Th sign test is a binomial test with p=0.5.

binom.test(x=8,
           n=9,
           p=0.5,
           alternative = "g")

    Exact binomial test

data:  8 and 9
number of successes = 8, number of trials = 9, p-value = 0.01953
alternative hypothesis: true probability of success is greater than 0.5
95 percent confidence interval:
 0.5708645 1.0000000
sample estimates:
probability of success 
             0.8888889 

Example 5:

• 6 students went on a diet to lose weight. A Pair is (WB, WA)
  • 5 lost weight (WB > WA) ==> -
  • 1 gained weight (WB < WA) ==> +
• Is the diet effective?
• To answer this we run an lower tailed sign test.

Th sign test is a binomial test with p=0.5.

binom.test(x=1,
           n=6,
           p=0.5,
           alternative = "l")

    Exact binomial test

data:  1 and 6
number of successes = 1, number of trials = 6, p-value = 0.1094
alternative hypothesis: true probability of success is less than 0.5
95 percent confidence interval:
 0.0000000 0.5818034
sample estimates:
probability of success 
             0.1666667 

Tolerance Limits

Example 6:

• Electric seat adjusters.
• A car manufacture wants to know how much range of adjustment is needed to be 90% sure that at least 80% of the population of buyers will be able to adjust their seats.

#install.packages("tolerance") # install package
library(tolerance)

# How to find Q
distfree.est(n = 18, # sample size
             alpha = 0.1, # 1-alpha is the confidence
             P= NULL,
             side = 2. # two sided interval
               )

        18
0.1 0.8006

# How to find n
distfree.est(n = NULL, # sample size
             alpha = 0.1, # 1-alpha is the confidence
             P= 0.8,
             side = 2. # two sided interval
               )

    0.8
0.1  18