Consider the problem below:

3 out of 100 New Yorkers have red hair. Construct a 90% confidence interval for the proportion of red haired people in the State of New York.

### Step 1 – Calculate the Sample Proportion

**The Sample Proportion** is the part of a given sample of individuals in larger group that follows a specific characteristic.

The parameter p′ represents the sample proportion and is used to calculate the genuine population proportion.

- The following equation is used to find the sample proportion:

**p′=x/n**

**p′**= the sample proportion**x**= the number of the sample population that meets the criteria**n**= the whole number of the sample population

**Sample Proportion Calculation:**

**x **= 3; **n= **100

Now, **p′ = x/n**

⇒ **p′ = **3/100

⇒ **p′ = 0.03**

### Step 2 – Calculate the Margin of Error

The** Margin of Error** is a value that estimates the percentage of sample points that can differ from real values.

- The formula to calculate the Margin of Error is:

**E**= Margin of Error**Z**= Critical Value_{c}**p′=**the sample proportion**n=**the whole number of the sample population

**Critical Values** represent the rejection region of a hypothesis test.

The following image shows critical values for different confidence intervals.

- To choose the correct Zc value from the list, use the following formula in C6.

`=INDEX('Sample Data'!B11:C15,MATCH(C5,'Sample Data'!B11:B15,0),2)`

- Calculate the Margin of Error by using the following formula:

`=C5*SQRT(C4*(1-C4)/C6)`

### Step 3 – Calculate the Population Proportion

- Taking the Margin of Error into account, the following equation will represent the population proportion with a confidence interval:

**(p′-E)< P <(p′+E)**

- As the Margin of Error is 0.0281, we get the left and right boundary of population proportion as shown below:

**p′- E = **0.03 + 0.0281 =0.0581

**p′+ E= **0.03 – 0.0281=0.0019

From the above inequality, we can conclude that the population proportion of redheads among the people in the New York State will be between 5.81% to 0.19%.

The normal distribution is shown in the image below. The population proportion will range between 0.19% to 5.81% for a confidence level of 90%. So, the range of population proportion for our sample problem will be:

**0.0019 < P < 0.0581 **

Or,

**0.19% < P < 5.81%**

**Read More:** How to Calculate Confidence Interval for Population Mean in Excel

**Download Practice Workbook**

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