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STATISTICS &
     PROBABILITY
      Z-test on Proportion
Chi-square test on K-proportions
Z-test on Proportion & Chi-
       square test on K-proportions
   Test of hypotheses concerning proportions are
             important in many areas.
            Some of which are as follows:

• Suppose the production manager wants to know if there is
  evidence of improvement in the production processes
  by testing if the proportion or the number of defectives
  had been reduced.
Z-test on Proportion & Chi-
      square test on K-proportions
• Hotel owners are interested in knowing if there is a
  significant difference between the proportion of
  clients who are satisfied and not satisfied in the
  services provided by the hotel personnel.

• A candidate would want to know her/his chances of
  winning by finding out if the proportion of people who
  will vote for her/him is significantly higher than those
  who are most likely to vote for her/his opponents.
Z-test on Proportion & Chi-
       square test on K-proportions
 • The discipline officer wants to know if the number new
   policy on smoking has reduced the number of smokers
   in CSB.

• The guidance counselor wants to know if the number of
  female dress code violators is significantly higher than
  the number of male dress code violator.
Z-test on Proportion & Chi-
square test on K-proportions
Z-test on Proportion
   For testing the significance of difference between
       sample proportion and hypothesized value

    x − npo
Z=                           x = the number of successes
   npo (1 − po )             n= the number of sample
                             Po = the hypothesized proportion


           Using PHStat: Go to…
  “One-Sample Tests; Z-Test on Proportions…”
Z-test on Proportion
  For testing the significance ofGo to…
                  Using PHStat: difference between
“Two-Sample Tests; Z-Test for Differences in Two Prop…”
                two sample proportions
   Or Chi-Square Test for Differences in Two Prop…
              x1 x2
                −                       x1 + x2
              n1 n2             and p =
Z=
                   1       1          n1 + n2
         p (1 − p ) + 
                   n         
                    1 n2 
    x1 = the number of successes in the first group
    x2 = the number of successes in the second group
    n1 = the number of sample in the first group
    n2 = the number of sample in the second group
TESTING A CLAIM ABOUT STANDARD
  DEVIATION OR VARIANCE
Test Statistics for Testing Hypothesis About σ or σ2 :
Chi-square distribution:


               x   2
                       =
                         ( n −1) s   2

                            σ   2

    where: n = sample size
           s2 = sample variance
           σ2 = population variance (given in the null
                hypothesis
Criterion:

    1.One − tailed left directional
                     2    2
    Reject H 0 if, x ≤   x 1−α
    2.One − tailed right directional
                     2     2
    Reject H 0 if, x ≥ x       α

    3.Two − tailed
                     2    2 α         2      2α
    Reject H 0 if, x ≤   x 1− and     x ≥x
                             2                2
Chi-square test on two or
             K-Proportions

            X =∑
              2        ( o − e)   2


                           e
where: “o” stands for observed frequencies
  and “e” stands for the expected frequencies.

             Using PHStat: Go to…
   “Multiple-Sample Tests; Chi-Square Test”
EXERCISES: Test Concerning Proportions
1. At a certain college, it is estimated that 25%of
    the students have cars on campus. Does this
    seem to be valid estimate if, in a random
    sample of 90 college students, 28 are found to
    have cars. Use a 0.05 level of confidence.
 2. A cigarette-manufacturing firm distributes two
    brands of cigarettes. It is found that 56 of 200
    smokers prefer brand A and that 29 of 150
    smokers prefer brand B, can we conclude at
    0.01 level of significance that brand A outsells
    brand B?
3. It has been claimed that 55% of students dislike
  mathematics. When a survey as conducted, it
  showed that 153 of 600 students dislike
  mathematics. Test if the claim is too high at
   α = 0.05 .
4. In a factory of baby dresses, one production
  process yielded 30 defective pieces in a random
  sample of 400, while another yielded 17
  defective pieces in a random sample of 300. Is
  there a significant difference between the
  proportions of defective baby dresses? Test at
  0.01 level of significance.
5. The Office of the Dean will conduct a research
 on the study grants recipients in a certain
 school. It was reported that 25% of the present
 grantees are ineligible and should not have
 been receiving any grants. You are hired to
 investigate the claim and in a survey you
 conducted, you found that out of 200 grantees,
 30 should have been disqualified. At α = 0.05
 level, should the Dean’s Office claim be
 rejected?
6. With individual lines at its various windows, the
  CSB bank found that the standard deviation for
  normally distributed waiting lines on Friday
  afternoons was 6.2 minutes. The bank
  experimented with a single main waiting line and
  found that for that for random sample of 45
  customers, the waiting times have a standard
  deviation of 3.8 mins. Based on previous
  studies, we can assume that the waiting lines
  are normally distributed. At α = 0.05, test the
  claim that a single line causes lower variation
  among the lines.
7. In a study of the wide ranges in the
  academic success of college freshmen,
  one various factor is the amount spend in
  studying. At the 0.01 level of significance,
  test the claim that the standard deviation
  is more than 4 hours. The sample consists
  of 70 randomly selected freshmen who
  have a standard deviation of 5.33 hours.
8. ABS Corporation have been successfully
  manufacturing electronic parts with a
  standard deviation of 43.7 from the
  existing line. After the installation of the
  new line a sample of 50 products was
  inspected from the line and found that the
  standard deviation was 54.7. Has the
  standard deviation of the total products
  changed with the new equipment? Use
  α = 0.05.

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Z-test & Chi-square Tests for Proportions & Variances

  • 1. STATISTICS & PROBABILITY Z-test on Proportion Chi-square test on K-proportions
  • 2. Z-test on Proportion & Chi- square test on K-proportions Test of hypotheses concerning proportions are important in many areas. Some of which are as follows: • Suppose the production manager wants to know if there is evidence of improvement in the production processes by testing if the proportion or the number of defectives had been reduced.
  • 3. Z-test on Proportion & Chi- square test on K-proportions • Hotel owners are interested in knowing if there is a significant difference between the proportion of clients who are satisfied and not satisfied in the services provided by the hotel personnel. • A candidate would want to know her/his chances of winning by finding out if the proportion of people who will vote for her/him is significantly higher than those who are most likely to vote for her/his opponents.
  • 4. Z-test on Proportion & Chi- square test on K-proportions • The discipline officer wants to know if the number new policy on smoking has reduced the number of smokers in CSB. • The guidance counselor wants to know if the number of female dress code violators is significantly higher than the number of male dress code violator.
  • 5. Z-test on Proportion & Chi- square test on K-proportions
  • 6. Z-test on Proportion For testing the significance of difference between sample proportion and hypothesized value x − npo Z= x = the number of successes npo (1 − po ) n= the number of sample Po = the hypothesized proportion Using PHStat: Go to… “One-Sample Tests; Z-Test on Proportions…”
  • 7. Z-test on Proportion For testing the significance ofGo to… Using PHStat: difference between “Two-Sample Tests; Z-Test for Differences in Two Prop…” two sample proportions Or Chi-Square Test for Differences in Two Prop… x1 x2 − x1 + x2 n1 n2 and p = Z= 1 1  n1 + n2 p (1 − p ) +  n   1 n2  x1 = the number of successes in the first group x2 = the number of successes in the second group n1 = the number of sample in the first group n2 = the number of sample in the second group
  • 8. TESTING A CLAIM ABOUT STANDARD DEVIATION OR VARIANCE Test Statistics for Testing Hypothesis About σ or σ2 : Chi-square distribution: x 2 = ( n −1) s 2 σ 2 where: n = sample size s2 = sample variance σ2 = population variance (given in the null hypothesis
  • 9. Criterion: 1.One − tailed left directional 2 2 Reject H 0 if, x ≤ x 1−α 2.One − tailed right directional 2 2 Reject H 0 if, x ≥ x α 3.Two − tailed 2 2 α 2 2α Reject H 0 if, x ≤ x 1− and x ≥x 2 2
  • 10. Chi-square test on two or K-Proportions X =∑ 2 ( o − e) 2 e where: “o” stands for observed frequencies and “e” stands for the expected frequencies. Using PHStat: Go to… “Multiple-Sample Tests; Chi-Square Test”
  • 11. EXERCISES: Test Concerning Proportions 1. At a certain college, it is estimated that 25%of the students have cars on campus. Does this seem to be valid estimate if, in a random sample of 90 college students, 28 are found to have cars. Use a 0.05 level of confidence. 2. A cigarette-manufacturing firm distributes two brands of cigarettes. It is found that 56 of 200 smokers prefer brand A and that 29 of 150 smokers prefer brand B, can we conclude at 0.01 level of significance that brand A outsells brand B?
  • 12. 3. It has been claimed that 55% of students dislike mathematics. When a survey as conducted, it showed that 153 of 600 students dislike mathematics. Test if the claim is too high at α = 0.05 . 4. In a factory of baby dresses, one production process yielded 30 defective pieces in a random sample of 400, while another yielded 17 defective pieces in a random sample of 300. Is there a significant difference between the proportions of defective baby dresses? Test at 0.01 level of significance.
  • 13. 5. The Office of the Dean will conduct a research on the study grants recipients in a certain school. It was reported that 25% of the present grantees are ineligible and should not have been receiving any grants. You are hired to investigate the claim and in a survey you conducted, you found that out of 200 grantees, 30 should have been disqualified. At α = 0.05 level, should the Dean’s Office claim be rejected?
  • 14. 6. With individual lines at its various windows, the CSB bank found that the standard deviation for normally distributed waiting lines on Friday afternoons was 6.2 minutes. The bank experimented with a single main waiting line and found that for that for random sample of 45 customers, the waiting times have a standard deviation of 3.8 mins. Based on previous studies, we can assume that the waiting lines are normally distributed. At α = 0.05, test the claim that a single line causes lower variation among the lines.
  • 15. 7. In a study of the wide ranges in the academic success of college freshmen, one various factor is the amount spend in studying. At the 0.01 level of significance, test the claim that the standard deviation is more than 4 hours. The sample consists of 70 randomly selected freshmen who have a standard deviation of 5.33 hours.
  • 16. 8. ABS Corporation have been successfully manufacturing electronic parts with a standard deviation of 43.7 from the existing line. After the installation of the new line a sample of 50 products was inspected from the line and found that the standard deviation was 54.7. Has the standard deviation of the total products changed with the new equipment? Use α = 0.05.