Large Sample Size
Small Sample Size
Hypothesis Tests for Population
Proportions
Bernd Schröder
Bernd Schröder
Hypothesis Tests for Population Proportions
logo1
Louisiana Tech University, College of Engineering and Science
Large Sample Size
Small Sample Size
To test H0 : p = p0 ...
Bernd Schröder
Hypothesis Tests for Population Proportions
logo1
Louisiana Tech University, College of Engineering and Science
Large Sample Size
Small Sample Size
To test H0 : p = p0 ...
... as long as np0 ≥ 10 and n(1 − p0 ) ≥ 10.
Bernd Schröder
Hypothesis Tests for Population Proportions
logo1
Louisiana Tech University, College of Engineering and Science
Large Sample Size
Small Sample Size
To test H0 : p = p0 ...
... as long as np0 ≥ 10 and n(1 − p0 ) ≥ 10. (So that the normal
approximation for the binomial distribution works.)
Bernd Schröder
Hypothesis Tests for Population Proportions
logo1
Louisiana Tech University, College of Engineering and Science
Large Sample Size
Small Sample Size
To test H0 : p = p0 ...
... as long as np0 ≥ 10 and n(1 − p0 ) ≥ 10. (So that the normal
approximation for the binomial distribution works.)
p̂ − p0
1. Test statistic: z = p
p0 (1 − p0 )/n
Bernd Schröder
Hypothesis Tests for Population Proportions
logo1
Louisiana Tech University, College of Engineering and Science
Large Sample Size
Small Sample Size
To test H0 : p = p0 ...
... as long as np0 ≥ 10 and n(1 − p0 ) ≥ 10. (So that the normal
approximation for the binomial distribution works.)
p̂ − p0
1. Test statistic: z = p
p0 (1 − p0 )/n
2. Alternative hypotheses and rejection regions.
Bernd Schröder
Hypothesis Tests for Population Proportions
logo1
Louisiana Tech University, College of Engineering and Science
Large Sample Size
Small Sample Size
To test H0 : p = p0 ...
... as long as np0 ≥ 10 and n(1 − p0 ) ≥ 10. (So that the normal
approximation for the binomial distribution works.)
p̂ − p0
1. Test statistic: z = p
p0 (1 − p0 )/n
2. Alternative hypotheses and rejection regions.
I
For Ha : p > p0 use z ≥ zα (upper tailed test)
Bernd Schröder
Hypothesis Tests for Population Proportions
logo1
Louisiana Tech University, College of Engineering and Science
Large Sample Size
Small Sample Size
To test H0 : p = p0 ...
... as long as np0 ≥ 10 and n(1 − p0 ) ≥ 10. (So that the normal
approximation for the binomial distribution works.)
p̂ − p0
1. Test statistic: z = p
p0 (1 − p0 )/n
2. Alternative hypotheses and rejection regions.
I
For Ha : p > p0 use z ≥ zα (upper tailed test)
-
Bernd Schröder
Hypothesis Tests for Population Proportions
logo1
Louisiana Tech University, College of Engineering and Science
Large Sample Size
Small Sample Size
To test H0 : p = p0 ...
... as long as np0 ≥ 10 and n(1 − p0 ) ≥ 10. (So that the normal
approximation for the binomial distribution works.)
p̂ − p0
1. Test statistic: z = p
p0 (1 − p0 )/n
2. Alternative hypotheses and rejection regions.
I
For Ha : p > p0 use z ≥ zα (upper tailed test)
-
Bernd Schröder
Hypothesis Tests for Population Proportions
logo1
Louisiana Tech University, College of Engineering and Science
Large Sample Size
Small Sample Size
To test H0 : p = p0 ...
... as long as np0 ≥ 10 and n(1 − p0 ) ≥ 10. (So that the normal
approximation for the binomial distribution works.)
p̂ − p0
1. Test statistic: z = p
p0 (1 − p0 )/n
2. Alternative hypotheses and rejection regions.
I
For Ha : p > p0 use z ≥ zα (upper tailed test)
µ0
Bernd Schröder
Hypothesis Tests for Population Proportions
logo1
Louisiana Tech University, College of Engineering and Science
Large Sample Size
Small Sample Size
To test H0 : p = p0 ...
... as long as np0 ≥ 10 and n(1 − p0 ) ≥ 10. (So that the normal
approximation for the binomial distribution works.)
p̂ − p0
1. Test statistic: z = p
p0 (1 − p0 )/n
2. Alternative hypotheses and rejection regions.
I
For Ha : p > p0 use z ≥ zα (upper tailed test)
µ0
Bernd Schröder
Hypothesis Tests for Population Proportions
logo1
Louisiana Tech University, College of Engineering and Science
Large Sample Size
Small Sample Size
To test H0 : p = p0 ...
... as long as np0 ≥ 10 and n(1 − p0 ) ≥ 10. (So that the normal
approximation for the binomial distribution works.)
p̂ − p0
1. Test statistic: z = p
p0 (1 − p0 )/n
2. Alternative hypotheses and rejection regions.
I
For Ha : p > p0 use z ≥ zα (upper tailed test)
α
µ0
Bernd Schröder
Hypothesis Tests for Population Proportions
logo1
Louisiana Tech University, College of Engineering and Science
Large Sample Size
Small Sample Size
To test H0 : p = p0 ...
... as long as np0 ≥ 10 and n(1 − p0 ) ≥ 10. (So that the normal
approximation for the binomial distribution works.)
p̂ − p0
1. Test statistic: z = p
p0 (1 − p0 )/n
2. Alternative hypotheses and rejection regions.
I
For Ha : p > p0 use z ≥ zα (upper tailed test)
α
µ0
Upper tail test for µ≤µ0 :
Bernd Schröder
Hypothesis Tests for Population Proportions
logo1
Louisiana Tech University, College of Engineering and Science
Large Sample Size
Small Sample Size
To test H0 : p = p0 ...
... as long as np0 ≥ 10 and n(1 − p0 ) ≥ 10. (So that the normal
approximation for the binomial distribution works.)
p̂ − p0
1. Test statistic: z = p
p0 (1 − p0 )/n
2. Alternative hypotheses and rejection regions.
I
For Ha : p > p0 use z ≥ zα (upper tailed test)
α
µ0
Upper tail test for µ≤µ0 :
Tail probability is ≤α (small) if µ≤µ0 .
Bernd Schröder
Hypothesis Tests for Population Proportions
logo1
Louisiana Tech University, College of Engineering and Science
Large Sample Size
Small Sample Size
To test H0 : p = p0 ...
... as long as np0 ≥ 10 and n(1 − p0 ) ≥ 10. (So that the normal
approximation for the binomial distribution works.)
p̂ − p0
1. Test statistic: z = p
p0 (1 − p0 )/n
2. Alternative hypotheses and rejection regions.
I
For Ha : p > p0 use z ≥ zα (upper tailed test)
Bernd Schröder
Hypothesis Tests for Population Proportions
logo1
Louisiana Tech University, College of Engineering and Science
Large Sample Size
Small Sample Size
To test H0 : p = p0 ...
... as long as np0 ≥ 10 and n(1 − p0 ) ≥ 10. (So that the normal
approximation for the binomial distribution works.)
p̂ − p0
1. Test statistic: z = p
p0 (1 − p0 )/n
2. Alternative hypotheses and rejection regions.
I
I
For Ha : p > p0 use z ≥ zα (upper tailed test)
For Ha : p < p0 use z ≤ −zα (lower tailed test)
Bernd Schröder
Hypothesis Tests for Population Proportions
logo1
Louisiana Tech University, College of Engineering and Science
Large Sample Size
Small Sample Size
To test H0 : p = p0 ...
... as long as np0 ≥ 10 and n(1 − p0 ) ≥ 10. (So that the normal
approximation for the binomial distribution works.)
p̂ − p0
1. Test statistic: z = p
p0 (1 − p0 )/n
2. Alternative hypotheses and rejection regions.
I
I
For Ha : p > p0 use z ≥ zα (upper tailed test)
For Ha : p < p0 use z ≤ −zα (lower tailed test)
-
Bernd Schröder
Hypothesis Tests for Population Proportions
logo1
Louisiana Tech University, College of Engineering and Science
Large Sample Size
Small Sample Size
To test H0 : p = p0 ...
... as long as np0 ≥ 10 and n(1 − p0 ) ≥ 10. (So that the normal
approximation for the binomial distribution works.)
p̂ − p0
1. Test statistic: z = p
p0 (1 − p0 )/n
2. Alternative hypotheses and rejection regions.
I
I
For Ha : p > p0 use z ≥ zα (upper tailed test)
For Ha : p < p0 use z ≤ −zα (lower tailed test)
-
Bernd Schröder
Hypothesis Tests for Population Proportions
logo1
Louisiana Tech University, College of Engineering and Science
Large Sample Size
Small Sample Size
To test H0 : p = p0 ...
... as long as np0 ≥ 10 and n(1 − p0 ) ≥ 10. (So that the normal
approximation for the binomial distribution works.)
p̂ − p0
1. Test statistic: z = p
p0 (1 − p0 )/n
2. Alternative hypotheses and rejection regions.
I
I
For Ha : p > p0 use z ≥ zα (upper tailed test)
For Ha : p < p0 use z ≤ −zα (lower tailed test)
µ0
Bernd Schröder
Hypothesis Tests for Population Proportions
logo1
Louisiana Tech University, College of Engineering and Science
Large Sample Size
Small Sample Size
To test H0 : p = p0 ...
... as long as np0 ≥ 10 and n(1 − p0 ) ≥ 10. (So that the normal
approximation for the binomial distribution works.)
p̂ − p0
1. Test statistic: z = p
p0 (1 − p0 )/n
2. Alternative hypotheses and rejection regions.
I
I
For Ha : p > p0 use z ≥ zα (upper tailed test)
For Ha : p < p0 use z ≤ −zα (lower tailed test)
µ0
Bernd Schröder
Hypothesis Tests for Population Proportions
logo1
Louisiana Tech University, College of Engineering and Science
Large Sample Size
Small Sample Size
To test H0 : p = p0 ...
... as long as np0 ≥ 10 and n(1 − p0 ) ≥ 10. (So that the normal
approximation for the binomial distribution works.)
p̂ − p0
1. Test statistic: z = p
p0 (1 − p0 )/n
2. Alternative hypotheses and rejection regions.
I
I
For Ha : p > p0 use z ≥ zα (upper tailed test)
For Ha : p < p0 use z ≤ −zα (lower tailed test)
α
µ0
Bernd Schröder
Hypothesis Tests for Population Proportions
logo1
Louisiana Tech University, College of Engineering and Science
Large Sample Size
Small Sample Size
To test H0 : p = p0 ...
... as long as np0 ≥ 10 and n(1 − p0 ) ≥ 10. (So that the normal
approximation for the binomial distribution works.)
p̂ − p0
1. Test statistic: z = p
p0 (1 − p0 )/n
2. Alternative hypotheses and rejection regions.
I
I
For Ha : p > p0 use z ≥ zα (upper tailed test)
For Ha : p < p0 use z ≤ −zα (lower tailed test)
α
µ0
Lower tail test for µ≥µ0 :
Bernd Schröder
Hypothesis Tests for Population Proportions
logo1
Louisiana Tech University, College of Engineering and Science
Large Sample Size
Small Sample Size
To test H0 : p = p0 ...
... as long as np0 ≥ 10 and n(1 − p0 ) ≥ 10. (So that the normal
approximation for the binomial distribution works.)
p̂ − p0
1. Test statistic: z = p
p0 (1 − p0 )/n
2. Alternative hypotheses and rejection regions.
I
I
For Ha : p > p0 use z ≥ zα (upper tailed test)
For Ha : p < p0 use z ≤ −zα (lower tailed test)
α
µ0
Lower tail test for µ≥µ0 :
Tail probability is ≤α (small) if µ≥µ0 .
Bernd Schröder
Hypothesis Tests for Population Proportions
logo1
Louisiana Tech University, College of Engineering and Science
Large Sample Size
Small Sample Size
To test H0 : p = p0 ...
... as long as np0 ≥ 10 and n(1 − p0 ) ≥ 10. (So that the normal
approximation for the binomial distribution works.)
p̂ − p0
1. Test statistic: z = p
p0 (1 − p0 )/n
2. Alternative hypotheses and rejection regions.
I
I
For Ha : p > p0 use z ≥ zα (upper tailed test)
For Ha : p < p0 use z ≤ −zα (lower tailed test)
Bernd Schröder
Hypothesis Tests for Population Proportions
logo1
Louisiana Tech University, College of Engineering and Science
Large Sample Size
Small Sample Size
To test H0 : p = p0 ...
... as long as np0 ≥ 10 and n(1 − p0 ) ≥ 10. (So that the normal
approximation for the binomial distribution works.)
p̂ − p0
1. Test statistic: z = p
p0 (1 − p0 )/n
2. Alternative hypotheses and rejection regions.
I
I
I
For Ha : p > p0 use z ≥ zα (upper tailed test)
For Ha : p < p0 use z ≤ −zα (lower tailed test)
For Ha : p 6= p0 use |z| ≥ z α2 (two-tailed test)
Bernd Schröder
Hypothesis Tests for Population Proportions
logo1
Louisiana Tech University, College of Engineering and Science
Large Sample Size
Small Sample Size
To test H0 : p = p0 ...
... as long as np0 ≥ 10 and n(1 − p0 ) ≥ 10. (So that the normal
approximation for the binomial distribution works.)
p̂ − p0
1. Test statistic: z = p
p0 (1 − p0 )/n
2. Alternative hypotheses and rejection regions.
I
I
I
For Ha : p > p0 use z ≥ zα (upper tailed test)
For Ha : p < p0 use z ≤ −zα (lower tailed test)
For Ha : p 6= p0 use |z| ≥ z α2 (two-tailed test)
-
Bernd Schröder
Hypothesis Tests for Population Proportions
logo1
Louisiana Tech University, College of Engineering and Science
Large Sample Size
Small Sample Size
To test H0 : p = p0 ...
... as long as np0 ≥ 10 and n(1 − p0 ) ≥ 10. (So that the normal
approximation for the binomial distribution works.)
p̂ − p0
1. Test statistic: z = p
p0 (1 − p0 )/n
2. Alternative hypotheses and rejection regions.
I
I
I
For Ha : p > p0 use z ≥ zα (upper tailed test)
For Ha : p < p0 use z ≤ −zα (lower tailed test)
For Ha : p 6= p0 use |z| ≥ z α2 (two-tailed test)
-
Bernd Schröder
Hypothesis Tests for Population Proportions
logo1
Louisiana Tech University, College of Engineering and Science
Large Sample Size
Small Sample Size
To test H0 : p = p0 ...
... as long as np0 ≥ 10 and n(1 − p0 ) ≥ 10. (So that the normal
approximation for the binomial distribution works.)
p̂ − p0
1. Test statistic: z = p
p0 (1 − p0 )/n
2. Alternative hypotheses and rejection regions.
I
I
I
For Ha : p > p0 use z ≥ zα (upper tailed test)
For Ha : p < p0 use z ≤ −zα (lower tailed test)
For Ha : p 6= p0 use |z| ≥ z α2 (two-tailed test)
µ0
Bernd Schröder
Hypothesis Tests for Population Proportions
logo1
Louisiana Tech University, College of Engineering and Science
Large Sample Size
Small Sample Size
To test H0 : p = p0 ...
... as long as np0 ≥ 10 and n(1 − p0 ) ≥ 10. (So that the normal
approximation for the binomial distribution works.)
p̂ − p0
1. Test statistic: z = p
p0 (1 − p0 )/n
2. Alternative hypotheses and rejection regions.
I
I
I
For Ha : p > p0 use z ≥ zα (upper tailed test)
For Ha : p < p0 use z ≤ −zα (lower tailed test)
For Ha : p 6= p0 use |z| ≥ z α2 (two-tailed test)
µ0
Bernd Schröder
Hypothesis Tests for Population Proportions
logo1
Louisiana Tech University, College of Engineering and Science
Large Sample Size
Small Sample Size
To test H0 : p = p0 ...
... as long as np0 ≥ 10 and n(1 − p0 ) ≥ 10. (So that the normal
approximation for the binomial distribution works.)
p̂ − p0
1. Test statistic: z = p
p0 (1 − p0 )/n
2. Alternative hypotheses and rejection regions.
I
I
I
For Ha : p > p0 use z ≥ zα (upper tailed test)
For Ha : p < p0 use z ≤ −zα (lower tailed test)
For Ha : p 6= p0 use |z| ≥ z α2 (two-tailed test)
α
2
µ0
Bernd Schröder
Hypothesis Tests for Population Proportions
logo1
Louisiana Tech University, College of Engineering and Science
Large Sample Size
Small Sample Size
To test H0 : p = p0 ...
... as long as np0 ≥ 10 and n(1 − p0 ) ≥ 10. (So that the normal
approximation for the binomial distribution works.)
p̂ − p0
1. Test statistic: z = p
p0 (1 − p0 )/n
2. Alternative hypotheses and rejection regions.
I
I
I
For Ha : p > p0 use z ≥ zα (upper tailed test)
For Ha : p < p0 use z ≤ −zα (lower tailed test)
For Ha : p 6= p0 use |z| ≥ z α2 (two-tailed test)
α
2
µ0
Bernd Schröder
Hypothesis Tests for Population Proportions
logo1
Louisiana Tech University, College of Engineering and Science
Large Sample Size
Small Sample Size
To test H0 : p = p0 ...
... as long as np0 ≥ 10 and n(1 − p0 ) ≥ 10. (So that the normal
approximation for the binomial distribution works.)
p̂ − p0
1. Test statistic: z = p
p0 (1 − p0 )/n
2. Alternative hypotheses and rejection regions.
I
I
I
For Ha : p > p0 use z ≥ zα (upper tailed test)
For Ha : p < p0 use z ≤ −zα (lower tailed test)
For Ha : p 6= p0 use |z| ≥ z α2 (two-tailed test)
α
2
α
2
µ0
Bernd Schröder
Hypothesis Tests for Population Proportions
logo1
Louisiana Tech University, College of Engineering and Science
Large Sample Size
Small Sample Size
To test H0 : p = p0 ...
... as long as np0 ≥ 10 and n(1 − p0 ) ≥ 10. (So that the normal
approximation for the binomial distribution works.)
p̂ − p0
1. Test statistic: z = p
p0 (1 − p0 )/n
2. Alternative hypotheses and rejection regions.
I
I
I
For Ha : p > p0 use z ≥ zα (upper tailed test)
For Ha : p < p0 use z ≤ −zα (lower tailed test)
For Ha : p 6= p0 use |z| ≥ z α2 (two-tailed test)
α
2
α
2
µ0
Two tailed test for µ 6= µ0 :
Bernd Schröder
Hypothesis Tests for Population Proportions
logo1
Louisiana Tech University, College of Engineering and Science
Large Sample Size
Small Sample Size
To test H0 : p = p0 ...
... as long as np0 ≥ 10 and n(1 − p0 ) ≥ 10. (So that the normal
approximation for the binomial distribution works.)
p̂ − p0
1. Test statistic: z = p
p0 (1 − p0 )/n
2. Alternative hypotheses and rejection regions.
I
I
I
For Ha : p > p0 use z ≥ zα (upper tailed test)
For Ha : p < p0 use z ≤ −zα (lower tailed test)
For Ha : p 6= p0 use |z| ≥ z α2 (two-tailed test)
α
2
α
2
µ0
Two tailed test for µ 6= µ0 :
logo1
Tail probability is α (small) if µ=µ0Louisiana
.
Bernd Schröder
Tech University, College of Engineering and Science
Hypothesis Tests for Population Proportions
Large Sample Size
Small Sample Size
Example.
Bernd Schröder
Hypothesis Tests for Population Proportions
logo1
Louisiana Tech University, College of Engineering and Science
Large Sample Size
Small Sample Size
Example. An airline claims that, on average, 5% of its flights
are delayed each day. On a given day, of 500 flights, 50 are
delayed. Test the hypothesis that the average proportion of
delayed flights is 5% at the 0.01 level.
Bernd Schröder
Hypothesis Tests for Population Proportions
logo1
Louisiana Tech University, College of Engineering and Science
Large Sample Size
Small Sample Size
Example. An airline claims that, on average, 5% of its flights
are delayed each day. On a given day, of 500 flights, 50 are
delayed. Test the hypothesis that the average proportion of
delayed flights is 5% at the 0.01 level.
Parameter of interest:
Bernd Schröder
Hypothesis Tests for Population Proportions
logo1
Louisiana Tech University, College of Engineering and Science
Large Sample Size
Small Sample Size
Example. An airline claims that, on average, 5% of its flights
are delayed each day. On a given day, of 500 flights, 50 are
delayed. Test the hypothesis that the average proportion of
delayed flights is 5% at the 0.01 level.
Parameter of interest: p0 , average proportion of delayed flights.
Bernd Schröder
Hypothesis Tests for Population Proportions
logo1
Louisiana Tech University, College of Engineering and Science
Large Sample Size
Small Sample Size
Example. An airline claims that, on average, 5% of its flights
are delayed each day. On a given day, of 500 flights, 50 are
delayed. Test the hypothesis that the average proportion of
delayed flights is 5% at the 0.01 level.
Parameter of interest: p0 , average proportion of delayed flights.
Null hypothesis:
Bernd Schröder
Hypothesis Tests for Population Proportions
logo1
Louisiana Tech University, College of Engineering and Science
Large Sample Size
Small Sample Size
Example. An airline claims that, on average, 5% of its flights
are delayed each day. On a given day, of 500 flights, 50 are
delayed. Test the hypothesis that the average proportion of
delayed flights is 5% at the 0.01 level.
Parameter of interest: p0 , average proportion of delayed flights.
Null hypothesis: H0 : p0 = 5% = 0.05
Bernd Schröder
Hypothesis Tests for Population Proportions
logo1
Louisiana Tech University, College of Engineering and Science
Large Sample Size
Small Sample Size
Example. An airline claims that, on average, 5% of its flights
are delayed each day. On a given day, of 500 flights, 50 are
delayed. Test the hypothesis that the average proportion of
delayed flights is 5% at the 0.01 level.
Parameter of interest: p0 , average proportion of delayed flights.
Null hypothesis: H0 : p0 = 5% = 0.05
Alternative hypothesis:
Bernd Schröder
Hypothesis Tests for Population Proportions
logo1
Louisiana Tech University, College of Engineering and Science
Large Sample Size
Small Sample Size
Example. An airline claims that, on average, 5% of its flights
are delayed each day. On a given day, of 500 flights, 50 are
delayed. Test the hypothesis that the average proportion of
delayed flights is 5% at the 0.01 level.
Parameter of interest: p0 , average proportion of delayed flights.
Null hypothesis: H0 : p0 = 5% = 0.05
Alternative hypothesis: Ha : p0 > 0.05
Bernd Schröder
Hypothesis Tests for Population Proportions
logo1
Louisiana Tech University, College of Engineering and Science
Large Sample Size
Small Sample Size
Example. An airline claims that, on average, 5% of its flights
are delayed each day. On a given day, of 500 flights, 50 are
delayed. Test the hypothesis that the average proportion of
delayed flights is 5% at the 0.01 level.
Parameter of interest: p0 , average proportion of delayed flights.
Null hypothesis: H0 : p0 = 5% = 0.05
Alternative hypothesis: Ha : p0 > 0.05 (the main concern is a
larger average proportion than what is claimed).
Bernd Schröder
Hypothesis Tests for Population Proportions
logo1
Louisiana Tech University, College of Engineering and Science
Large Sample Size
Small Sample Size
Example. An airline claims that, on average, 5% of its flights
are delayed each day. On a given day, of 500 flights, 50 are
delayed. Test the hypothesis that the average proportion of
delayed flights is 5% at the 0.01 level.
Parameter of interest: p0 , average proportion of delayed flights.
Null hypothesis: H0 : p0 = 5% = 0.05
Alternative hypothesis: Ha : p0 > 0.05 (the main concern is a
larger average proportion than what is claimed).
Test statistic:
Bernd Schröder
Hypothesis Tests for Population Proportions
logo1
Louisiana Tech University, College of Engineering and Science
Large Sample Size
Small Sample Size
Example. An airline claims that, on average, 5% of its flights
are delayed each day. On a given day, of 500 flights, 50 are
delayed. Test the hypothesis that the average proportion of
delayed flights is 5% at the 0.01 level.
Parameter of interest: p0 , average proportion of delayed flights.
Null hypothesis: H0 : p0 = 5% = 0.05
Alternative hypothesis: Ha : p0 > 0.05 (the main concern is a
larger average proportion than what is claimed).
p̂ − p0
Test statistic: z = p
p0 (1 − p0 )/n
Bernd Schröder
Hypothesis Tests for Population Proportions
logo1
Louisiana Tech University, College of Engineering and Science
Large Sample Size
Small Sample Size
Example. An airline claims that, on average, 5% of its flights
are delayed each day. On a given day, of 500 flights, 50 are
delayed. Test the hypothesis that the average proportion of
delayed flights is 5% at the 0.01 level.
Parameter of interest: p0 , average proportion of delayed flights.
Null hypothesis: H0 : p0 = 5% = 0.05
Alternative hypothesis: Ha : p0 > 0.05 (the main concern is a
larger average proportion than what is claimed).
p̂ − p0
Test statistic: z = p
p0 (1 − p0 )/n
Rejection region:
Bernd Schröder
Hypothesis Tests for Population Proportions
logo1
Louisiana Tech University, College of Engineering and Science
Large Sample Size
Small Sample Size
Example. An airline claims that, on average, 5% of its flights
are delayed each day. On a given day, of 500 flights, 50 are
delayed. Test the hypothesis that the average proportion of
delayed flights is 5% at the 0.01 level.
Parameter of interest: p0 , average proportion of delayed flights.
Null hypothesis: H0 : p0 = 5% = 0.05
Alternative hypothesis: Ha : p0 > 0.05 (the main concern is a
larger average proportion than what is claimed).
p̂ − p0
Test statistic: z = p
p0 (1 − p0 )/n
Rejection region: z > z0.01 ≈ 2.33.
Bernd Schröder
Hypothesis Tests for Population Proportions
logo1
Louisiana Tech University, College of Engineering and Science
Large Sample Size
Small Sample Size
Example. An airline claims that, on average, 5% of its flights
are delayed each day. On a given day, of 500 flights, 50 are
delayed. Test the hypothesis that the average proportion of
delayed flights is 5% at the 0.01 level.
Parameter of interest: p0 , average proportion of delayed flights.
Null hypothesis: H0 : p0 = 5% = 0.05
Alternative hypothesis: Ha : p0 > 0.05 (the main concern is a
larger average proportion than what is claimed).
p̂ − p0
Test statistic: z = p
p0 (1 − p0 )/n
Rejection region: z > z0.01 ≈ 2.33.
Substitute values into test statistic:
Bernd Schröder
Hypothesis Tests for Population Proportions
logo1
Louisiana Tech University, College of Engineering and Science
Large Sample Size
Small Sample Size
Example. An airline claims that, on average, 5% of its flights
are delayed each day. On a given day, of 500 flights, 50 are
delayed. Test the hypothesis that the average proportion of
delayed flights is 5% at the 0.01 level.
Parameter of interest: p0 , average proportion of delayed flights.
Null hypothesis: H0 : p0 = 5% = 0.05
Alternative hypothesis: Ha : p0 > 0.05 (the main concern is a
larger average proportion than what is claimed).
p̂ − p0
Test statistic: z = p
p0 (1 − p0 )/n
Rejection region: z > z0.01 ≈ 2.33.
Substitute values into test statistic:
p̂ − p0
z= p
p0 (1 − p0 )/n
Bernd Schröder
Hypothesis Tests for Population Proportions
logo1
Louisiana Tech University, College of Engineering and Science
Large Sample Size
Small Sample Size
Example. An airline claims that, on average, 5% of its flights
are delayed each day. On a given day, of 500 flights, 50 are
delayed. Test the hypothesis that the average proportion of
delayed flights is 5% at the 0.01 level.
Parameter of interest: p0 , average proportion of delayed flights.
Null hypothesis: H0 : p0 = 5% = 0.05
Alternative hypothesis: Ha : p0 > 0.05 (the main concern is a
larger average proportion than what is claimed).
p̂ − p0
Test statistic: z = p
p0 (1 − p0 )/n
Rejection region: z > z0.01 ≈ 2.33.
Substitute values into test statistic:
50
− 0.05
p̂ − p0
500
z= p
=p
p0 (1 − p0 )/n
0.05(1 − 0.05)/500
Bernd Schröder
Hypothesis Tests for Population Proportions
logo1
Louisiana Tech University, College of Engineering and Science
Large Sample Size
Small Sample Size
Example. An airline claims that, on average, 5% of its flights
are delayed each day. On a given day, of 500 flights, 50 are
delayed. Test the hypothesis that the average proportion of
delayed flights is 5% at the 0.01 level.
Parameter of interest: p0 , average proportion of delayed flights.
Null hypothesis: H0 : p0 = 5% = 0.05
Alternative hypothesis: Ha : p0 > 0.05 (the main concern is a
larger average proportion than what is claimed).
p̂ − p0
Test statistic: z = p
p0 (1 − p0 )/n
Rejection region: z > z0.01 ≈ 2.33.
Substitute values into test statistic:
50
− 0.05
p̂ − p0
500
z= p
=p
≈ 5.13
p0 (1 − p0 )/n
0.05(1 − 0.05)/500
Bernd Schröder
Hypothesis Tests for Population Proportions
logo1
Louisiana Tech University, College of Engineering and Science
Large Sample Size
Small Sample Size
Example. An airline claims that, on average, 5% of its flights
are delayed each day. On a given day, of 500 flights, 50 are
delayed. Test the hypothesis that the average proportion of
delayed flights is 5% at the 0.01 level.
Parameter of interest: p0 , average proportion of delayed flights.
Null hypothesis: H0 : p0 = 5% = 0.05
Alternative hypothesis: Ha : p0 > 0.05 (the main concern is a
larger average proportion than what is claimed).
p̂ − p0
Test statistic: z = p
p0 (1 − p0 )/n
Rejection region: z > z0.01 ≈ 2.33.
Substitute values into test statistic:
50
− 0.05
p̂ − p0
500
z= p
=p
≈ 5.13
p0 (1 − p0 )/n
0.05(1 − 0.05)/500
Decide if to reject or not.
logo1
Bernd Schröder
Hypothesis Tests for Population Proportions
Louisiana Tech University, College of Engineering and Science
Large Sample Size
Small Sample Size
Example. An airline claims that, on average, 5% of its flights
are delayed each day. On a given day, of 500 flights, 50 are
delayed. Test the hypothesis that the average proportion of
delayed flights is 5% at the 0.01 level.
Parameter of interest: p0 , average proportion of delayed flights.
Null hypothesis: H0 : p0 = 5% = 0.05
Alternative hypothesis: Ha : p0 > 0.05 (the main concern is a
larger average proportion than what is claimed).
p̂ − p0
Test statistic: z = p
p0 (1 − p0 )/n
Rejection region: z > z0.01 ≈ 2.33.
Substitute values into test statistic:
50
− 0.05
p̂ − p0
500
z= p
=p
≈ 5.13
p0 (1 − p0 )/n
0.05(1 − 0.05)/500
Decide if to reject or not. Reject.
logo1
Bernd Schröder
Hypothesis Tests for Population Proportions
Louisiana Tech University, College of Engineering and Science
Large Sample Size
Small Sample Size
To Test H0 : p = p0 Versus Ha : p > p0 ...
Bernd Schröder
Hypothesis Tests for Population Proportions
logo1
Louisiana Tech University, College of Engineering and Science
Large Sample Size
Small Sample Size
To Test H0 : p = p0 Versus Ha : p > p0 ...
... for small sample size
Bernd Schröder
Hypothesis Tests for Population Proportions
logo1
Louisiana Tech University, College of Engineering and Science
Large Sample Size
Small Sample Size
To Test H0 : p = p0 Versus Ha : p > p0 ...
... for small sample size, use the binomial distribution directly.
Bernd Schröder
Hypothesis Tests for Population Proportions
logo1
Louisiana Tech University, College of Engineering and Science
Large Sample Size
Small Sample Size
To Test H0 : p = p0 Versus Ha : p > p0 ...
... for small sample size, use the binomial distribution directly.
1. Test statistic: Number x of favorable outcomes.
Bernd Schröder
Hypothesis Tests for Population Proportions
logo1
Louisiana Tech University, College of Engineering and Science
Large Sample Size
Small Sample Size
To Test H0 : p = p0 Versus Ha : p > p0 ...
... for small sample size, use the binomial distribution directly.
1. Test statistic: Number x of favorable outcomes.
2. Rejection region: x ≥ c, where c is an integer.
Bernd Schröder
Hypothesis Tests for Population Proportions
logo1
Louisiana Tech University, College of Engineering and Science
Large Sample Size
Small Sample Size
To Test H0 : p = p0 Versus Ha : p > p0 ...
... for small sample size, use the binomial distribution directly.
1. Test statistic: Number x of favorable outcomes.
2. Rejection region: x ≥ c, where c is an integer.
3. Type I errors.
Bernd Schröder
Hypothesis Tests for Population Proportions
logo1
Louisiana Tech University, College of Engineering and Science
Large Sample Size
Small Sample Size
To Test H0 : p = p0 Versus Ha : p > p0 ...
... for small sample size, use the binomial distribution directly.
1. Test statistic: Number x of favorable outcomes.
2. Rejection region: x ≥ c, where c is an integer.
3. Type I errors.
P(type I error)
Bernd Schröder
Hypothesis Tests for Population Proportions
logo1
Louisiana Tech University, College of Engineering and Science
Large Sample Size
Small Sample Size
To Test H0 : p = p0 Versus Ha : p > p0 ...
... for small sample size, use the binomial distribution directly.
1. Test statistic: Number x of favorable outcomes.
2. Rejection region: x ≥ c, where c is an integer.
3. Type I errors.
P(type I error) = P(X ≥ c|X ∼ Bin(n, p0 ))
Bernd Schröder
Hypothesis Tests for Population Proportions
logo1
Louisiana Tech University, College of Engineering and Science
Large Sample Size
Small Sample Size
To Test H0 : p = p0 Versus Ha : p > p0 ...
... for small sample size, use the binomial distribution directly.
1. Test statistic: Number x of favorable outcomes.
2. Rejection region: x ≥ c, where c is an integer.
3. Type I errors.
P(type I error) = P(X ≥ c|X ∼ Bin(n, p0 ))
= 1 − P(X < c|X ∼ Bin(n, p0 ))
Bernd Schröder
Hypothesis Tests for Population Proportions
logo1
Louisiana Tech University, College of Engineering and Science
Large Sample Size
Small Sample Size
To Test H0 : p = p0 Versus Ha : p > p0 ...
... for small sample size, use the binomial distribution directly.
1. Test statistic: Number x of favorable outcomes.
2. Rejection region: x ≥ c, where c is an integer.
3. Type I errors.
P(type I error) = P(X ≥ c|X ∼ Bin(n, p0 ))
= 1 − P(X < c|X ∼ Bin(n, p0 ))
= 1 − B(c − 1; n; p0 )
Bernd Schröder
Hypothesis Tests for Population Proportions
logo1
Louisiana Tech University, College of Engineering and Science
Large Sample Size
Small Sample Size
To Test H0 : p = p0 Versus Ha : p > p0 ...
... for small sample size, use the binomial distribution directly.
1. Test statistic: Number x of favorable outcomes.
2. Rejection region: x ≥ c, where c is an integer.
3. Type I errors.
P(type I error) = P(X ≥ c|X ∼ Bin(n, p0 ))
= 1 − P(X < c|X ∼ Bin(n, p0 ))
= 1 − B(c − 1; n; p0 )
4. Type II errors.
Bernd Schröder
Hypothesis Tests for Population Proportions
logo1
Louisiana Tech University, College of Engineering and Science
Large Sample Size
Small Sample Size
To Test H0 : p = p0 Versus Ha : p > p0 ...
... for small sample size, use the binomial distribution directly.
1. Test statistic: Number x of favorable outcomes.
2. Rejection region: x ≥ c, where c is an integer.
3. Type I errors.
P(type I error) = P(X ≥ c|X ∼ Bin(n, p0 ))
= 1 − P(X < c|X ∼ Bin(n, p0 ))
= 1 − B(c − 1; n; p0 )
4. Type II errors.
β (p0 )
Bernd Schröder
Hypothesis Tests for Population Proportions
logo1
Louisiana Tech University, College of Engineering and Science
Large Sample Size
Small Sample Size
To Test H0 : p = p0 Versus Ha : p > p0 ...
... for small sample size, use the binomial distribution directly.
1. Test statistic: Number x of favorable outcomes.
2. Rejection region: x ≥ c, where c is an integer.
3. Type I errors.
P(type I error) = P(X ≥ c|X ∼ Bin(n, p0 ))
= 1 − P(X < c|X ∼ Bin(n, p0 ))
= 1 − B(c − 1; n; p0 )
4. Type II errors.
β (p0 ) = P(X < c|X ∼ Bin(n, p0 ))
Bernd Schröder
Hypothesis Tests for Population Proportions
logo1
Louisiana Tech University, College of Engineering and Science
Large Sample Size
Small Sample Size
To Test H0 : p = p0 Versus Ha : p > p0 ...
... for small sample size, use the binomial distribution directly.
1. Test statistic: Number x of favorable outcomes.
2. Rejection region: x ≥ c, where c is an integer.
3. Type I errors.
P(type I error) = P(X ≥ c|X ∼ Bin(n, p0 ))
= 1 − P(X < c|X ∼ Bin(n, p0 ))
= 1 − B(c − 1; n; p0 )
4. Type II errors.
β (p0 ) = P(X < c|X ∼ Bin(n, p0 ))
= B(c − 1; n; p0 )
Bernd Schröder
Hypothesis Tests for Population Proportions
logo1
Louisiana Tech University, College of Engineering and Science
Large Sample Size
Small Sample Size
Example. An airline claims that on average 5% of its flights
are delayed each day.
Bernd Schröder
Hypothesis Tests for Population Proportions
logo1
Louisiana Tech University, College of Engineering and Science
Large Sample Size
Small Sample Size
Example. An airline claims that on average 5% of its flights
are delayed each day. On a given day, of 25 flights, 3 are
delayed.
Bernd Schröder
Hypothesis Tests for Population Proportions
logo1
Louisiana Tech University, College of Engineering and Science
Large Sample Size
Small Sample Size
Example. An airline claims that on average 5% of its flights
are delayed each day. On a given day, of 25 flights, 3 are
delayed. Test the hypothesis that the average proportion of
delayed flights is 5% at the 0.01 level.
Bernd Schröder
Hypothesis Tests for Population Proportions
logo1
Louisiana Tech University, College of Engineering and Science
Large Sample Size
Small Sample Size
Example. An airline claims that on average 5% of its flights
are delayed each day. On a given day, of 25 flights, 3 are
delayed. Test the hypothesis that the average proportion of
delayed flights is 5% at the 0.01 level.
α = 1 − B(c − 1; n; p0 )
Bernd Schröder
Hypothesis Tests for Population Proportions
logo1
Louisiana Tech University, College of Engineering and Science
Large Sample Size
Small Sample Size
Example. An airline claims that on average 5% of its flights
are delayed each day. On a given day, of 25 flights, 3 are
delayed. Test the hypothesis that the average proportion of
delayed flights is 5% at the 0.01 level.
α = 1 − B(c − 1; n; p0 )
0.01 = 1 − B(c − 1; 25; 0.05)
Bernd Schröder
Hypothesis Tests for Population Proportions
logo1
Louisiana Tech University, College of Engineering and Science
Large Sample Size
Small Sample Size
Example. An airline claims that on average 5% of its flights
are delayed each day. On a given day, of 25 flights, 3 are
delayed. Test the hypothesis that the average proportion of
delayed flights is 5% at the 0.01 level.
α = 1 − B(c − 1; n; p0 )
0.01 = 1 − B(c − 1; 25; 0.05)
B(c − 1; 25; 0.05) = 0.99
Bernd Schröder
Hypothesis Tests for Population Proportions
logo1
Louisiana Tech University, College of Engineering and Science
Large Sample Size
Small Sample Size
Example. An airline claims that on average 5% of its flights
are delayed each day. On a given day, of 25 flights, 3 are
delayed. Test the hypothesis that the average proportion of
delayed flights is 5% at the 0.01 level.
α
0.01
B(c − 1; 25; 0.05)
c−1
Bernd Schröder
Hypothesis Tests for Population Proportions
=
=
=
=
1 − B(c − 1; n; p0 )
1 − B(c − 1; 25; 0.05)
0.99
3
logo1
Louisiana Tech University, College of Engineering and Science
Large Sample Size
Small Sample Size
Example. An airline claims that on average 5% of its flights
are delayed each day. On a given day, of 25 flights, 3 are
delayed. Test the hypothesis that the average proportion of
delayed flights is 5% at the 0.01 level.
α
0.01
B(c − 1; 25; 0.05)
c−1
c
Bernd Schröder
Hypothesis Tests for Population Proportions
=
=
=
=
=
1 − B(c − 1; n; p0 )
1 − B(c − 1; 25; 0.05)
0.99
3
4
logo1
Louisiana Tech University, College of Engineering and Science
Large Sample Size
Small Sample Size
Example. An airline claims that on average 5% of its flights
are delayed each day. On a given day, of 25 flights, 3 are
delayed. Test the hypothesis that the average proportion of
delayed flights is 5% at the 0.01 level.
α
0.01
B(c − 1; 25; 0.05)
c−1
c
=
=
=
=
=
1 − B(c − 1; n; p0 )
1 − B(c − 1; 25; 0.05)
0.99
3
4
Conclusion: Do not reject.
Bernd Schröder
Hypothesis Tests for Population Proportions
logo1
Louisiana Tech University, College of Engineering and Science
Large Sample Size
Small Sample Size
Example. An airline claims that on average 5% of its flights
are delayed each day. On a given day, of 25 flights, 3 are
delayed.
Bernd Schröder
Hypothesis Tests for Population Proportions
logo1
Louisiana Tech University, College of Engineering and Science
Large Sample Size
Small Sample Size
Example. An airline claims that on average 5% of its flights
are delayed each day. On a given day, of 25 flights, 3 are
delayed. Compute the probability of a type II error when testing
the hypothesis that the average proportion of delayed flights is
5% at the 0.01 level and the actual proportion is 10%.
Bernd Schröder
Hypothesis Tests for Population Proportions
logo1
Louisiana Tech University, College of Engineering and Science
Large Sample Size
Small Sample Size
Example. An airline claims that on average 5% of its flights
are delayed each day. On a given day, of 25 flights, 3 are
delayed. Compute the probability of a type II error when testing
the hypothesis that the average proportion of delayed flights is
5% at the 0.01 level and the actual proportion is 10%.
β = B(c − 1; n; p0 )
Bernd Schröder
Hypothesis Tests for Population Proportions
logo1
Louisiana Tech University, College of Engineering and Science
Large Sample Size
Small Sample Size
Example. An airline claims that on average 5% of its flights
are delayed each day. On a given day, of 25 flights, 3 are
delayed. Compute the probability of a type II error when testing
the hypothesis that the average proportion of delayed flights is
5% at the 0.01 level and the actual proportion is 10%.
β = B(c − 1; n; p0 )
= B(3; 25; 0.1)
Bernd Schröder
Hypothesis Tests for Population Proportions
logo1
Louisiana Tech University, College of Engineering and Science
Large Sample Size
Small Sample Size
Example. An airline claims that on average 5% of its flights
are delayed each day. On a given day, of 25 flights, 3 are
delayed. Compute the probability of a type II error when testing
the hypothesis that the average proportion of delayed flights is
5% at the 0.01 level and the actual proportion is 10%.
β = B(c − 1; n; p0 )
= B(3; 25; 0.1)
= 0.764
Bernd Schröder
Hypothesis Tests for Population Proportions
logo1
Louisiana Tech University, College of Engineering and Science