POPULATION PROPORTION
Prepared by: Mr. Ian Anthony M. Torrente, LPT
𝑥−𝜇
𝑡 =
𝑠
√𝑛
𝐶𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛
: If n<30
𝑑𝑓
= ( 𝑛 −1 )
𝑧= 𝑥 − 𝜇
𝜎
√𝑛
𝐶𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛
: If n ≥30
𝐷𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑒
𝑡ℎ𝑒 𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑓𝑓 .
1. = 1.860
2. = -2.145
3. = 2.845
4. = -1.316
5. = 2.306
6. to the right.(one-tailed) =
7. to the left.( one-tailed) =
=
9. =
10. =
What is population proportion?
A population proportion is
a fraction of the
population that has a certain
characteristic.
A population proportion
generally denoted as P – is a
parameter that describes a
percentage value associated
with a population.
The formula for population proportion is given by:
¿ ¿
𝑝 −𝑝 𝑝 −𝑝
𝑧= 𝑜𝑟 𝑧=
𝑝𝑞 𝑝 (1 − 𝑝 )
√𝑛 √ 𝑛
𝐹𝑜𝑟𝑚𝑢𝑙𝑎
𝑖𝑓 𝑝¿ 𝑖𝑠 𝑚𝑖𝑠𝑠𝑖𝑛𝑔 :
¿ 𝑥
𝑝 =
• p = population proportion 𝑛
• n = sample size
• q = 1-p
• X = part of the sample
• N = total number of sample
The following steps when testing hypothesis
concerning a proportion:
Step 1 : State the null and alternative hypothesis
Step 2: Choose a level of significance and critical values.
Step 3: Compute for the test – statistics
Step 4: Decision
Step 5: Conclusion
0 0 0
Example 1:
The players believes the chance of winning in a game is greater than 65%. In a random sample
of 150 games , The players won 118 times. Is there enough evidence to suggest that the players
are cheaters? Use the 0.01 level of significance.
Step 1 : State the null and alternative hypothesis Step 4: Decision
𝐻
𝑜 : 𝑝 ≤65 % 𝐴𝑐𝑐𝑒𝑝𝑡 𝑡ℎ𝑒 𝑁𝑢𝑙𝑙 𝐻𝑦𝑝𝑜𝑡ℎ𝑒𝑠𝑖𝑠 , 𝑅𝑒𝑗𝑒𝑐𝑡𝐴𝑙𝑡𝑒𝑟𝑛𝑎𝑡𝑖𝑣𝑒 𝐻𝑦𝑝𝑜𝑡ℎ𝑒𝑠𝑖𝑠
𝐻
𝑎 : 𝑝>65 % (𝑐𝑙𝑎𝑖𝑚) 𝑅𝑒𝑗𝑒𝑐𝑡
𝑇ℎ𝑒 𝑁𝑢𝑙𝑙 𝐻𝑦𝑝𝑜𝑡ℎ𝑒𝑠𝑖𝑠 , 𝐴𝑐𝑐𝑒𝑝𝑡 𝑡ℎ𝑒 𝐴𝑙𝑡𝑒𝑟𝑛𝑎𝑡𝑖𝑣𝑒 𝐻𝑦𝑝𝑜𝑡ℎ𝑒𝑠𝑖𝑠
Step 2: Choose a level of significance and critical values.
∝=0.
01 Step 5: Conclusion
𝐶𝑟𝑖𝑡𝑖𝑐𝑎𝑙
𝑉𝑎𝑙𝑢𝑒=2. 326 𝑇ℎ𝑒
𝑐𝑙𝑎𝑖𝑚 𝑖𝑠𝑡𝑟𝑢𝑒
Step 3: Compute for the test – statistics
𝐺𝑖𝑣𝑒𝑛
: ¿
𝑝 =0 . 79 , 𝑝=0 . 65 , 𝑛=150 , 𝑞=0 .35
¿
𝑝 −𝑝 0 . 79 −0 . 65 0 . 14
𝑧= 𝑧= 𝑧=
𝑝 (𝑞) (0 . 65)(0 . 35) 0 . 038944404 𝑧=3
. 59
√ 𝑛 √ 150
Example 2: The NCHS claims that in 2002, 75% of children aged 2 to 17 saw a dentist in the past year. An
investigator wants to assess whether use of dental services is similar in children living in the city of Boston.
A sample of 125 children aged 2 to 17 living in Boston are surveyed and 64 reported seeing a dentist over
the past 12 months. Is there a significant difference in use of dental services between children living in
Boston and the national data? Use a level of significance of 0.05.
Step 1 : State the null and alternative hypothesis Step 4: Decision
𝐴𝑐𝑐𝑒𝑝𝑡 𝑡ℎ𝑒 𝑁𝑢𝑙𝑙 𝐻𝑦𝑝𝑜𝑡ℎ𝑒𝑠𝑖𝑠 , 𝑅𝑒𝑗𝑒𝑐𝑡 𝐴𝑙𝑡𝑒𝑟𝑛𝑎𝑡𝑖𝑣𝑒 𝐻𝑦𝑝𝑜𝑡ℎ𝑒𝑠𝑖𝑠
𝐻
𝑜 : 𝑝=75 %(𝑐𝑙𝑎𝑖𝑚)
𝐻
𝑎 : 𝑝 ≠7 5 %
𝑅𝑒𝑗𝑒𝑐𝑡
𝑇ℎ𝑒 𝑁𝑢𝑙𝑙 𝐻𝑦𝑝𝑜𝑡ℎ𝑒𝑠𝑖𝑠 , 𝐴𝑐𝑐𝑒𝑝𝑡 𝑡ℎ𝑒 𝐴𝑙𝑡𝑒𝑟𝑛𝑎𝑡𝑖𝑣𝑒 𝐻𝑦𝑝𝑜𝑡ℎ𝑒𝑠𝑖𝑠
Step 2: Choose a level of significance and critical values. Step 5: Conclusion
∝=0.
05 𝑇ℎ𝑒
𝑐𝑙𝑎𝑖𝑚 𝑖𝑠 𝑓𝑎𝑙𝑠𝑒 .
𝐶𝑟𝑖𝑡𝑖𝑐𝑎𝑙
𝑉𝑎𝑙𝑢𝑒=±1 . 960
Step 3: Compute for the test – statistics
𝐺𝑖𝑣𝑒𝑛
: 𝑝¿ =0 . 51, 𝑝=0 . 75 ,𝑛=125 , 𝑞=0. 25
𝑧= 𝑝
¿
−𝑝 0 . 51 −0 . 75 − 0 . 24 𝑧=− 6 .196 → −6 . 20
𝑧= 𝑧=
𝑝 (𝑞) (0 .75)(0 . 25) 0 . 038729833
√ 𝑛 √ 125
Example 3:
A research is conducted on a certain company believes last year showed that is less than or equal to 25% of
the employees would rather drink coffee than soft drinks during break time. The company has recently
decided to give free coffee during break time. In the new research conducted this year, out of 125 randomly
sampled employee 28% said that they would rather drinks coffee than soft drinks. At 0.05 level of
significance is there a sufficient evidence to suggest that the coffee drinkers have increased since the
company has decided to give free coffee during break time?
Step 1 : State the null and alternative hypothesis Step 4: Decision
𝐻
𝑜 : 𝑝 ≤25 % (𝑐𝑙𝑎𝑖𝑚 ) 𝐴𝑐𝑐𝑒𝑝𝑡 𝑡ℎ𝑒 𝑁𝑢𝑙𝑙 𝐻𝑦𝑝𝑜𝑡ℎ𝑒𝑠𝑖𝑠 , 𝑅𝑒𝑗𝑒𝑐𝑡 𝐴𝑙𝑡𝑒𝑟𝑛𝑎𝑡𝑖𝑣𝑒 𝐻𝑦𝑝𝑜𝑡ℎ𝑒𝑠𝑖𝑠
𝐻
𝑎 : 𝑝>25 % 𝑅𝑒𝑗𝑒𝑐𝑡
𝑇ℎ𝑒 𝑁𝑢𝑙𝑙 𝐻𝑦𝑝𝑜𝑡ℎ𝑒𝑠𝑖𝑠 , 𝐴𝑐𝑐𝑒𝑝𝑡 𝑡ℎ𝑒 𝐴𝑙𝑡𝑒𝑟𝑛𝑎𝑡𝑖𝑣𝑒 𝐻𝑦𝑝𝑜𝑡ℎ𝑒𝑠𝑖𝑠
Step 2: Choose a level of significance and critical values.
∝=0
. 05 Step 5: Conclusion
𝐶𝑟𝑖𝑡𝑖𝑐𝑎𝑙
𝑉𝑎𝑙𝑢𝑒=1. 645 𝑇ℎ𝑒
𝑐𝑙𝑎𝑖𝑚 𝑖𝑠𝑡𝑟𝑢𝑒
Step 3: Compute for the test – statistics
𝐺𝑖𝑣𝑒𝑛
: 𝑝¿ =0. 28 , 𝑝=0 . 25 , 𝑛=125 , 𝑞=0 . 75
¿
𝑝 −𝑝 0 . 28 −0 . 25 0 . 03 𝑧=0 . 7745 →0 . 77
𝑧= 𝑧= 𝑧=
𝑝 (𝑞) (0 . 25)(0 . 75) 0 . 038729833
√ 𝑛 √ 125
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AND
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