library(BSDA)
library(tidyverse)
library(ggpubr)
library(rstatix)
library(pander)

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Problem 1: Seven manufacturing companies agreed to implement a time management program in hopes of improving productivity. The average times, in minutes, it took the companies to produce the same quantity and kinds of part are listed on the table below. Does this information indicate that the program decreased the production time? Use a 0.05 level of significance and assume normal population distributions. (10 pts.)

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Solution:

  1. Statement of the Problem: Is the average time it took the companies to produce the same quantity and kinds of part before the time management program greater than the average time after?

    Let \(\mu_1\) be the average time it took the companies to produce the same quantity and kind of part before the program.
    Let \(\mu_2\) be the average time it took the companies to produce the same quantity and kind of part after the program.

  2. Ho: The average time it took the companies to produce the same quantity and kinds of part before the time management program is less than or equal to the average time after the time management program.

    Ha: The average time it took the companies to produce the same quantity and kinds of part before the time management program is greater than the average time after the time management program.

  3. Level of Significance: \(0.05\)

  4. Test Statistic: t statistic for paired samples

  5. Computation/Analysis using RStudio:

before <- c(75, 112, 89, 95, 80, 105, 110)
after <- c(70, 110, 88, 100, 80, 100, 99)
pander(t.test(before, after, paired = TRUE, mu=0, alternative = "greater"))
Paired t-test: before and after \(~\)
Test statistic df P value Alternative hypothesis mean of the differences
1.439 6 0.1001 greater 2.714
  1. Decision: Fail to reject Ho since the computed p-value is greater than the \(0.05\) significance level.
  2. Conclusion: There is sufficient data supporting the Ho. Hence, the data shows that the program has not significantly decreased production time.

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Problem2: The monthly returns in percentage of pesos of two investment portfolios were recorded for one year. Perform a hypothesis test at the 0.05 significance level to determine if there is sufficient evidence showing that there is no significant difference in the mean monthly percentage returns between the two investment portfolios. (10 pts.)

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Solution:

  1. Statement of the Problem: Is there a significant difference in the mean monthly returns between Portfolio I and Portfolio II?
    Let \(\mu_1\) be the mean monthly returns for Portfolio I.
    Let \(\mu_2\) be the mean monthly returns for Portfolio II.

  2. Ho: There is no significant difference in the mean monthly returns between Portfolio I and Portfolio II.
    Ha: There is a significant difference in the mean monthly returns between Portfolio I and Portfolio II.

  3. Level of Significance: \(0.05\)

  4. Test Statistic: t-statistic for independent samples

  5. Computation/Analysis:

port1 <- c(2.1, 1.2, -1.5, 1.9, 0.7, 2.5, 3.0, -2.2, 1.8, 0.5, 2.0, 1.5)
port2 <- c(2.9, 3.5, -2.8, 1.0, -3.0, 2.6, -3.5, 4.5, 1.5, 2.3, -1.0, 0.8)
var.test(port1, port2)
  
    F test to compare two variances
  
  data:  port1 and port2
  F = 0.3335, num df = 11, denom df = 11, p-value = 0.082
  alternative hypothesis: true ratio of variances is not equal to 1
  95 percent confidence interval:
   0.09600787 1.15848712
  sample estimates:
  ratio of variances 
           0.3335024

The population variances are equal \((p>0.05)\).

pander(t.test(port1, port2, mu = 0, alternative="two.sided", var.equal = TRUE))
Two Sample t-test: port1 and port2 \(~\)
Test statistic df P value Alternative hypothesis mean of x mean of y
0.4344 22 0.6683 two.sided 1.125 0.7333

  1. Decision: Fail to reject Ho since \(p > 0.05\).

  2. Conclusion: There is sufficient data supporting Ho. Hence, there is no significant difference in the mean monthly returns between Portfolio I and Portfolio II.

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Problem 3: The following data represents the semi-monthly salary of the faculty (in thousands) of four state universities. Faculties were randomly selected from each school. At the 5% level of significance, is there a significant difference among the salary of the faculties of the four state universities? (10 pts.)

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Solution:

  1. Statement of the Problem: Is there a significant difference among the mean salary of the faculties of the four state universities?

  2. Ho: There is no significant difference among the mean salary of the faculties of the four state universities.
    Ha: At least two stat universities significantly differ in terms of the mean salary fo the faculty

  3. Level of Significance: \(0.05\)

  4. Test Statistic: F-statistic

  5. Computations/Analysis:

salary <- c(15, 20, 16, 13, 17, 12, 19, 18, 10, 20, 23, 18, 16, 30, 15, 17, 16, 12)
univ <- c("A", "A", "A", "A", "A", "B", "B", "B", "B", "C", "C", "C", "C", "C", "D", "D", "D", "D")

data <- data.frame(univ, salary)
data
     univ salary
  1     A     15
  2     A     20
  3     A     16
  4     A     13
  5     A     17
  6     B     12
  7     B     19
  8     B     18
  9     B     10
  10    C     20
  11    C     23
  12    C     18
  13    C     16
  14    C     30
  15    D     15
  16    D     17
  17    D     16
  18    D     12
class(data$univ)
  [1] "factor"
class(data$salary)
  [1] "numeric"
ggboxplot(data, x="univ", y="salary")

data%>%group_by(univ)%>%identify_outliers(salary)
  [1] univ       salary     is.outlier is.extreme
  <0 rows> (or 0-length row.names)
# Homogeneity of variances:
levene_test(data, salary~univ)
  # A tibble: 1 x 4
      df1   df2 statistic     p
    <int> <int>     <dbl> <dbl>
  1     3    14      1.24 0.333
# Normality of residuals
anovamodel<-lm(salary~univ, data)
ggqqplot(residuals(anovamodel))

shapiro.test(residuals((anovamodel)))
  
    Shapiro-Wilk normality test
  
  data:  residuals((anovamodel))
  W = 0.96518, p-value = 0.7037

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There are no outliers; the popn variances are homogeneous \((p>0.05)\); the residuals are approximately normally distributed \((p > 0.05)\).
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pander(anova(anovamodel))
Analysis of Variance Table \(~\)
  Df Sum Sq Mean Sq F value Pr(>F)
univ 3 136.2 45.4 2.905 0.07184
Residuals 14 218.8 15.63 NA NA
  1. Decision: Fail to reject Ho since \(p > 0.05\).
  2. Conclusion: There is sufficient data supporting Ho. Hence, there is no significant difference in the mean monthly salary among the four state universities.