POPULATION
PROPORTION
DEFINITION OF TERMS
Proportion – a percentage of the population that exhibits the behavior. It is
a fraction expression where the favorable response is in numerator and the
total number of respondents is in the denominator.
Population Proportion - a fraction of the population that has a certain
characteristic.
POINT ESTIMATE FOR THE POPULATION
PROPORTION P
• It is a fraction expression where the favorable response is in numerator and the total
number of respondents is in the denominator.
• Formula:
Where:
𝒏 = number of observations in a simple random population
p = sample proportion of success (p hat)
q = failures (q hat)
• In a survey of 400 millennials, 129 like to watch marvel movies on the big
screen. Estimate the true proportion p and q where p ̂ is the proportion
of those who like to watch marvel movies on the big screen based on the
sample.
• Given: 𝑛 = 400; 𝑋 =129
• In a professional satisfaction survey in a certain company, 800 professionals were asked if
they are satisfied with their jobs. There were 523 who responded with Yes. What is the
proportion of this? What proportion responded with a No
• Given: 𝑛 = 800; X = 523
Thus, the proportion of Yes = 65.38% and the proportion of No = 34.63%.
INTERVAL ESTIMATE OF POPULATION
PROPORTION P
• It is a range of values that we expect to capture the population parameter. Confidence
Level for P proportion is the times we expect where does the population parameter is
located.
STEPS IN SOLVING FOR INTERVAL ESTIMATE OF
POPULATION PROPORTION P
1. Given Data
2. Critical Value (based on 𝒛∝/𝟐 from the confidence level) - refer to the T-distribution table
second to the last row, the row containing z at the bottom of the table
3. Margin of Error (E) – answer can be rounded off four decimal places
4. Interval Estimate (p - 𝐸, p + 𝐸)
• A recent survey of 1000 men players revealed that 57% of them are having fun in playing new release games of
the year. Using a 95% confidence interval to estimate the true proportion of men players who enjoy playing new
release games of the year.
• Step 1: Given Data
• 𝒏 = 𝟏𝟎𝟎𝟎 𝐩̂ = 𝟓𝟕% = 𝟎.𝟓7
• 𝐪 ̂ =1−p ̂ =1−0.57=𝟎.𝟒𝟑
• C𝒐𝒏𝒇𝒊𝒅𝒆𝒏𝒄𝒆 𝒍𝒆𝒗𝒆𝒍 = 𝟗𝟓%
Given: n = 1000; p = 57%; q = 43%
• Given: n = 1000; p = 57%; q = 43%
• 𝒛∝/𝟐 = 1.96
• E = .0307
• Step 4: (p ̂ − 𝐸, p ̂ +𝐸)
• = (0.57 − 0.0307,0.57 + 0.0307) =(0.5393,0.6007)
Thus, we are 95% confident that the population proportion lies between 53.93% and 60.07%
A random sample of 260 voters were asked if they planned to vote in the next midterm elections. 60 out of 260
voters said yes. Using these results, construct the true population proportion by 93% level of confidence.
Step 2: Since the level of confidence is not indicated on T-distribution table, we will be going to use the Standard
Normal Distribution table and use these steps:
a. 𝛼 = 1−𝑐𝑜𝑛𝑓𝑖𝑑𝑒𝑛𝑐𝑒 𝑙𝑒𝑣𝑒𝑙
b. 𝛼/2
c. Locate 𝛼 2 on the table of Standard Normal distribution to get the Z-score(take the absolute value of the
located Z-score).
Step 2:
𝛼 = 1−𝑐𝑜𝑛𝑓𝑖𝑑𝑒𝑛𝑐𝑒 𝑙𝑒𝑣𝑒𝑙
𝛼 = 1 – 0.93 = .07
𝛼/2
.07/2 = 0.0035
a. Locate 𝛼 2 on the table of Standard Normal distribution to get the Z-score(take the absolute value of the
located Z-score).
Step 2:
𝛼 = .07/2 = 0.0035
Locate 𝛼 2 on the table of Standard Normal distribution to get the Z-score(take the absolute value of the located Z-
score).
Step 2:
𝛼 = .07/2 = 0.0035
z∝/2 = 𝟑.𝟑𝟗
