One-Sample Hypothesis Testing - Proportions

before you start

This lesson builds on key concepts from an Introduction to Statistics course. Specifically, this tutorial assumes familiarity with the foundational idea of one-sample hypothesis tests for population proportions.

1. Introduction to One-Sample Hypothesis Testing for a Proportion

Hypothesis testing is a method used to make decisions or draw conclusions about a population based on sample data. In the context of hypothesis testing for proportions, the focus is on the percentage of a certain characteristic within a group. For instance, if a company wants to determine if more than 50% of its customers are satisfied with their service, hypothesis testing can be employed. This process involves setting up a hypothesis, collecting data, and using statistical methods to analyze whether the data supports the hypothesis.

In a similar fashion as in the last tutorial, the steps to perform a hypothesis test are as follows:

1. State the Hypotheses: Clearly define the null and alternative hypotheses.
2. Gather the Data: Gather data in a way that is designed to test the hypotheses.
3. Choose the Significance Level: Decide on the α level (for example, 0.05 or 0.10).
4. Perform a Statistical Test: Use an appropriate statistical test to analyze the data. You will use a z-test for hypothesis testing for proportions.
5. Make a Decision: Based on the test results, decide whether to reject or fail to reject the null hypothesis. This decision is guided by a p-value, which indicates the probability of observing the data if the null hypothesis is true.
6. Interpret the Results: Explain the results of the hypothesis test in the context of the business problem.

1a. Types of Hypothesis Tests for Proportions

Just like with population means, there are three types of hypothesis tests for population proportions, based on the direction of the test. A description of each type of test is provided below.

1. Two-Tailed Test:

• Purpose: To determine if the sample proportion is significantly different from the hypothesized population proportion, either higher or lower.
• Hypotheses:
  • Null Hypothesis H₀: (the population proportion is equal to a hypothesized population proportion)
  • Alternative Hypothesis H₁: (the population proportion is not equal to a hypothesized population proportion)

EXAMPLE

A company wants to test if the proportion of satisfied customers is different from 50%.

2. Right-Sided (One-Tailed) Test:

• Purpose: To determine if the sample proportion is significantly greater than the hypothesized population proportion.
• Hypotheses:
  • Null Hypothesis H₀: (the population proportion is equal to a hypothesized population proportion)
  • Alternative Hypothesis H₁: (the population proportion is greater than a hypothesized population proportion)

EXAMPLE

A company wants to test if more than 60% of its customers are satisfied with their service.

3. Left-Sided (One-Tailed) Test:

• Purpose: To determine if the sample proportion is significantly less than the hypothesized population proportion.
• Hypotheses:
  • Null Hypothesis H₀: (the population proportion is equal to a hypothesized population proportion)
  • Alternative Hypothesis H₁: (the population proportion is less than a hypothesized population proportion)

EXAMPLE

A company wants to test if less than 40% of its customers are dissatisfied with their service.

2. Hypothesis Testing for Proportions: Two-Tailed, Right-Tailed, and Left-Tailed Tests

In the upcoming sections, you will explore the different types of hypothesis tests used in hypothesis tests for proportions: two-tailed, right-tailed, and left-tailed tests. You will discover the purpose of each test, how to perform the test, and how to interpret the results. Whether you are analyzing financial metrics or customer satisfaction surveys, mastering these tests will enhance your analytical skills and decision-making capabilities.

2a. Two-Tailed Hypothesis Test for a Proportion

Let us walk through a practical example of performing a two-tailed hypothesis test for a proportion.

EXAMPLE

SavvyShoppers, a retail company, has recently launched a new customer loyalty program aimed at increasing customer retention and sales. The program offers various incentives, such as discounts, exclusive offers, and reward points for frequent purchases. The company wants to evaluate whether the new loyalty program has significantly changed the customer conversion rate compared to the historical conversion rate.

You need to conduct a two-tailed hypothesis test to determine if the new loyalty program has had a significant impact on the customer conversion rate (percentage of customers who make purchases after a new loyalty program is implemented).

In performing this hypothesis test, you will complete the following steps:

Step 1: State the Hypotheses

• H₀: (conversion rate is equal to the historical conversion rate of the hypothesized proportion, )
• H₁: (conversion rate is not equal to the historical conversion rate of )

Step 2: Gather the Data

In Excel, you have a sample of 100 customers that indicates whether the customer converted or not (that is, made a purchase or not) after the loyalty program was introduced. Using the data in Excel, you find:

• (the sample proportion of customers who converted (made a purchase) after the loyalty program was implemented)
• 

Step 3: Choose a Significance Level

Use a level of significance,

Step 4: Perform a Statistical Test

For hypothesis tests for proportions, you use a test statistic from a standard normal distribution, a z-test statistic.

where is the hypothesized proportion in the null hypothesis, which is 0.50, and is the sample proportion, which is 0.44 in this example. The test statistic tells you how many standard deviations the sample proportion is from the hypothesized proportion of 0.50.

A test statistic of -1.2 means that the sample proportion is 1.2 standard deviations below the hypothesized population proportion.

Just like for hypothesis tests for the mean, the test statistic directly relates back to the sampling distribution of the proportion. The green histogram below illustrates the sampling distribution of showing how the proportions of buyers from numerous samples of 100 people each form a distribution. The distribution is centered around the hypothesized population proportion, and has a standard error calculated using:

The blue dashed line at 0.44 represents the sample proportion of 0.44. The area under the distribution to the right of the blue dashed line represents the p-value, which is the probability of obtaining a test statistic as extreme as, or more extreme than, the observed test statistic of -1.2, if the null hypothesis were true.



The standard normal distribution is used to standardize the sample proportion. The test statistic (z) represents a specific point on the standard normal distribution. It shows how many standard deviations the sample proportion, is from the hypothesized population proportion, You use the standard normal distribution to find the p-value for the hypothesis test. The graph below shows the standard normal distribution for a two-tailed hypothesis test. The orange dashed lines at and show the observed test statistics. The orange shaded areas represent the p-value, indicating the probability of observing test statistics as extreme as or more extreme, under the null hypothesis.



In a two-tailed test, you look for extreme values on both ends of the distribution, because you want to see if the sample proportion is significantly different from the hypothesized proportion, either higher or lower.

So, you have two critical values:

• One at for the high end.
• One at for the low end.

You will now use Excel to find the p-value for this two-tailed hypothesis test, using the data in the conversion_rate.xlsx file. The column Converted in the data represents if a particular customer made a purchase after the loyalty program was implemented; 1 means a purchase was made and 0 means no purchase was made.

Perform the following.

1. In cell C3, enter sample size. In cell D3, enter 100.

2. In cell C4, enter sample proportion (p-hat). In cell D4, enter the following formula:

=COUNTIF(A2:A101,1)/COUNT(A2:A101)

This formula calculates the proportion (or percentage) of cells in the range A2:A101 that contain the value 1.

3. In cell C6, enter standard error. In cell D6, enter the following formula:

=SQRT((D2*(1-D2))/D3)

4. In cell C7, enter z test statistic. In cell D7, enter the following formula:

=(D4-D2)/D6

5. In cell C8, enter absolute value z test statistic. In cell D8, enter the following formula:

=ABS(D7)

6. In cell C9, enter p-value (two-tailed test). In cell D9, enter the following formula:

=2 * (1 - NORM.S.DIST(D8, TRUE))

In cell D8, you should obtain a p-value of 0.2301.

Let’s Break Down the Excel P-Value Calculation

Let’s explain the components of this Excel formula for finding the p-value.

The NORM.S.DIST() function in Excel is used to work with the standard normal distribution, which is a special case of the normal distribution with a mean of 0 and a standard deviation of 1. The NORM.S.DIST() function returns the probability that a standard normal random variable is less than or equal to a given value (z). This is useful for finding probabilities and p-values in hypothesis testing.

The syntax for the function is:

NORM.S.DIST(z, cumulative) where:

• z: The z-test statistic (or z-score) for which you want to find the probability. The z-value represents the number of standard deviations the sample proportion is from the hypothesized proportion, in the context of hypothesis testing for proportions.
• cumulative: A logical value (TRUE or FALSE) that determines the type of probability you want to calculate. When you set cumulative to TRUE, the function calculates the probability of observing a test statistic less than the observed z-test statistic. This means it gives you the total area under the curve to the left of the z-test statistic. Essentially, it tells you the likelihood of observing a value up to and including the z-test statistic.

For guidance, the Excel formulas are shown in the screenshot below.



Now, let’s break down the components of the Excel formula used to calculate the two-sided p-value piece by piece.

=2 * (1 - NORM.S.DIST(1.2, TRUE))

• NORM.S.DIST(1.2, TRUE):
  • Purpose: This part of the formula calculates the probability that a standard normal variable (z) is less than or equal to 1.2.
  • Explanation: It gives the area under the standard normal curve to the left of the z-test statistic 1.2. Essentially, it tells us how much of the distribution lies to the left of 1.2.
    
    
• 1 - NORM.S.DIST(1.2, TRUE):
  • Purpose: This part calculates the probability that a standard normal variable (z) is greater than 1.2.
  • Explanation: Since the total area under the curve is 1 (representing 100% probability), subtracting the area to the left of 1.2 from 1 gives us the area to the right of 1.2.
    
    
• 2 * (1 - NORM.S.DIST(1.2, TRUE)):
  • Purpose: This part calculates the p-value for a two-tailed test.
  • Explanation: In a two-tailed test, you are interested in extreme values on both ends of the distribution. Therefore, you double the area in one tail to account for both tails. This gives the total probability of observing a value as extreme as 1.2, or more extreme, in either direction (both positive and negative).

Using Excel, you find the p-value to be 0.2301.

Step 5: Make a Decision

Since the p-value (level of significance), you fail to reject the null hypothesis.

Step 6: Interpret the Results

Since the p-value (0.2301) is greater than the significance level (0.05), you fail to reject the null hypothesis. There is not enough evidence to conclude that the new loyalty program has significantly changed the customer conversion rate from the historical conversion rate of 50%. 

try it

A bank has recently introduced a new premium banking service aimed at attracting more customers. Historically, the opt-in rate for new services has been 40%. The bank wants to evaluate whether the new premium service has significantly changed the opt-in rate.

The customer_opt_in_banking_service.xlsx Excel file contains a column named Opted_In, which indicates whether a customer opted in for the new premium banking service or not. A 1 means the customer opted in for the new service, and a 0 means the customer did not opt in for the new service.

Using the data in the customer_opt_in_banking_service.xlsx Excel file, conduct a hypothesis test to determine if the opt-in rate for the new banking service is significantly different from the historical rate of 40%. Use a significance level of 0.05. Interpret the results of the hypothesis test.

Solution:



Null and alternative hypotheses:

• H₀:
• H₁:

Using Excel, you find the p-value for this test to be 0.0433. The Excel worksheets with the values of the test statistic and p-value are provided below.



Interpretation: Since the p-value (0.0433) is less than the significance level (0.05), you reject the null hypothesis. Based on the sample of customers, there is sufficient statistical evidence to conclude that the opt-in rate for the new banking service is significantly different from the historical rate of 40%. Therefore, we can infer that the new premium banking service has had a significant impact on the customer opt-in rate.

watch

Check out this video on conducting a two-tailed hypothesis test to analyze the opt-in rate.

2b. Excel NORM.S.DIST() Functions for Calculating P-Values

This table provides a guide on which Excel NORM.S.DIST() function to use for calculating p-values in different types of z-tests. It includes functions for two-tailed, right-tailed, and left-tailed tests, specifying the appropriate function and a brief description of each. The value of z in the table represents the test statistic.

2c. Right-Tailed Hypothesis Test for a Proportion

Let’s walk through a practical example of a right-tailed hypothesis test for a proportion.

EXAMPLE

A university wants to determine if the new online class format has significantly increased student preference for online classes, compared to a historical preference rate of 55%. After implementing the new online class format, the university conducted a survey of 100 students to see if the preference rate has increased.

You need to conduct a right-tailed hypothesis test to determine if the new online class format has significantly increased student preference for online classes, compared to the historical preference rate of 55%. This test will help you evaluate whether the proportion of students who prefer online classes is greater than 55% after the implementation of the new format.

In performing this hypothesis test, you will complete the following steps.

Step 1: State the Hypotheses

• H₀: (student online preference rate for online classes is equal to the historical preference rate of the hypothesized proportion, )
• H₁: (student online preference rate is not equal to the historical online preference rate of )

Step 2: Gather the Data

In Excel, you have a sample of 100 student responses from a survey that indicates whether the student prefers an online or in-person format class. Using the data in Excel, you find:

• (the sample proportion of students who prefer the online class format)
• 

Step 3: Choose a Significance Level

Use a level of significance,

Step 4: Perform a Statistical Test

For hypothesis tests for proportions, you use a test statistic from a standard normal distribution, a z-test statistic.

where is the hypothesized proportion in the null hypothesis, which is 0.55, and is the sample proportion, which is 0.65 in this example. The test statistic tells you how many standard deviations the sample proportion is from the hypothesized proportion of 0.55.

A test statistic of 2 means that the sample proportion is 2 standard deviations above the hypothesized population proportion.

You use the standard normal distribution to find the p-value for the right-tailed hypothesis test. The graph below shows the standard normal distribution for a right-tailed hypothesis test. The orange dashed line at shows the observed test statistic. The orange shaded area represents the p-value, indicating the probability of observing the test statistic, or something more extreme, if the null hypothesis is true.



In a right-tailed test, you look for extreme values to the right end of the distribution, because you want to see if the sample proportion is significantly greater than the hypothesized mean.

You will now use Excel to find the p-value for this right-tailed hypothesis test, using the data in the student_preferences.xlsx file.

For guidance, the Excel formulas are shown in the screenshot below.



Perform the following.

1. In cell C2, enter hypothesized proportion. In cell D2, enter 0.55.

2. In cell C3, enter sample size. In cell D3, enter 100.

3. In cell C4, enter sample proportion (p-hat). In cell D4, enter the following formula:

=COUNTIF(A2:A101,”Online”)/COUNTA(A2:A101)

This formula calculates the proportion (or percentage) of cells in the range A2:A101 that contain the value “Online.”

4. Notice that the denominator for this sample proportion is using the Excel function COUNTA() and not COUNT(). COUNT() is used to count numeric values. COUNTA() is used to count any data type in an Excel cell. Since the data type in the Preference column is text, COUNTA() is the appropriate function to use to count the number of students that prefer the online format.

5. In cell C6, enter standard error. In cell D6, enter the following formula:

=SQRT((D2*(1-D2))/D3)

6. In cell C7, enter z test statistic. In cell D7, enter the following formula:

=(D4-D2)/D6

7. In cell C8, enter p-value (right-tailed test). In cell D8, enter the following formula:

=1-NORM.S.DIST(D7,TRUE)

Your Excel spreadsheet should contain these values for all of the computations.



Step 5: Make a Decision

Since the p-value (level of significance), you reject the null hypothesis.

Step 6: Interpret the Results

Since the p-value (0.0222) is less than the significance level (0.05), you reject the null hypothesis. This means there is sufficient evidence to conclude that the proportion of students who prefer online classes is significantly greater than the historical preference rate of 55%.

try it

A tech company wants to determine if a new customer service chatbot has significantly increased customer satisfaction compared to the previous quarter. The company surveyed 150 customers after implementing the new chatbot to see if the satisfaction rate had increased. The previous quarter’s satisfaction rate was 50% (that is, 50% of customers reported being satisfied after a customer service encounter).

The Excel sheet named customer_satisfaction_chatbot.xlsx contains survey data from 150 customers regarding their satisfaction with a new customer service chatbot. There are two columns in the data. Customer_ID is a unique identifier for each customer. Satisfaction is the satisfaction status of each customer with two possible values:

• Satisfied: Indicates customer was satisfied with the customer service provided by the new chatbot.
• Not Satisfied: Indicates customer was not satisfied with the customer service provided by the new chatbot.

Using the data in the customer_satisfaction_chatbot.xlsx Excel file, conduct a hypothesis test to determine if the new chatbot has significantly increased customer satisfaction compared to the previous quarter’s satisfaction rate of 50%. Use a significance level of 0.05. Interpret the results of the hypothesis test.

The Excel worksheets with the values of the test statistic and p-value are provided below.



Solution:



Null and alternative hypotheses:

• H₀:
• H₁:

Using Excel, you find the p-value for this test to be 0.0072.

Interpretation: Since the p-value (0.0072) is less than the significance level (0.05), you reject the null hypothesis. Based on the sample of customers, there is sufficient statistical evidence to conclude that the new customer service chatbot has increased customer satisfaction compared to the previous quarter’s satisfaction rate of 50%.

watch

Follow along with this video on conducting a right-tailed hypothesis test to determine chatbot impact on customer satisfaction.

2d. Left-Tailed Hypothesis Test for a Proportion

Let’s walk through a practical example of a left-tailed hypothesis test for a proportion.

EXAMPLE

You are working as a data analyst for BrightPay Credit, a credit card company. Historically, 80% of customers pay their credit card bills on time. Recently, there have been concerns that the proportion of on-time payments might have decreased. To investigate this claim, you take a sample of 100 customers and record whether each customer paid their bill on time. You need to conduct a left-tailed hypothesis test to determine if the company’s concern is valid.

In performing this hypothesis test, you will complete the following steps.

Step 1: State the Hypotheses

• H₀: (proportion of on-time payments is equal to the historical rate of the hypothesized proportion, )
• H₁: (proportion of on-time payments is less than the historical rate of )

Step 2: Gather the Data

In Excel, you have a sample of 100 customers and an indicator variable that denotes whether the customer has paid their bill on time or not for the last billing cycle. Using the data in Excel, you find:

• (the sample proportion of customers who have paid their bill on time)
• 

Step 3: Choose a Significance Level

Use a level of significance,

Step 4: Perform a Statistical Test

For hypothesis tests for proportions, you use a test statistic from a standard normal distribution, a z-test statistic.

where is the hypothesized proportion in the null hypothesis, which is 0.80, and is the sample proportion, which is 0.70 in this example. The test statistic tells you how many standard deviations the sample proportion is from the hypothesized proportion of 0.80.

A test statistic of -2.5 means that the sample proportion is 2.5 standard deviations below the hypothesized population proportion.

You use the standard normal distribution to find the p-value for the left-tailed hypothesis test. The graph below shows the standard normal distribution for a left-tailed hypothesis test. The orange dashed line at shows the observed test statistic. The orange shaded area represents the p-value, indicating the probability of observing the test statistic, or something more extreme, if the null hypothesis is true.



In a left-tailed test, you look for extreme values to the left end of the distribution, because you want to see if the sample proportion is significantly less than the hypothesized mean.

You will now use Excel to find the p-value for this left-tailed hypothesis test, using the data in the on_time_payments.xlsx file. A value of 1 in the OnTime_Payment column indicates that a customer paid on time. A value of 0 indicates that a customer paid late.

For guidance, the Excel formulas are shown in the screenshot below.



Perform the following.

1. In cell C2, enter hypothesized proportion. In cell D2, enter 0.80.

2. In cell C3, enter sample size. In cell D3, enter 100.

3. In cell C4, enter sample proportion (p-hat). In cell D4, enter the following formula:

=COUNTIF(A2:A101, 1)/COUNT(A2:A101)

This formula calculates the proportion (or percentage) of customers with on-time payments.

4. In cell C6, enter standard error. In cell D6, enter the following formula:

=SQRT((D2*(1-D2))/D3)

5. In cell C7, enter z test statistic. In cell D7, enter the following formula:

=(D4-D2)/D6

6. In cell C8, enter p-value (left-tailed test). In cell D8, enter the following formula:

=NORM.S.DIST(D7,TRUE)

Your Excel spreadsheet should contain these values for all of the computations.



Step 5: Make a Decision

Since the p-value (level of significance), you reject the null hypothesis.

Step 6: Interpret the Results

Since the p-value (0.0062) is less than the significance level (0.05), you reject the null hypothesis. This means there is evidence to conclude that the current proportion of on-time payments is significantly less than the historical rate of 80%.

try it

A financial firm wants to determine if a new investment strategy has significantly decreased the proportion of investments that meet or exceed the target return compared to the previous quarter. In the past, the proportion of investments that meet or exceed a target return has been 50%. The firm collected a sample of 200 investments after implementing the new strategy to see if the proportion had decreased from the historical rate of 50%.

The Excel sheet named investment_returns.xlsx contains a random sample of 200 investments regarding their performance relative to a target return. There are two columns in the data. Investment_ID is a unique identifier for each investment. Investment_Return is the performance status of each investment, with two possible values:

• Above Target: Investment met or exceeded the target return.
• Below Target: Investment did not meet the target return.

Using the data in the investment_return.xlsx Excel file, conduct a hypothesis test to determine if the proportion of investments meeting or exceeding the target return is less than 50%. Use a significance level of 0.05. Interpret the results of the hypothesis test.

Solution:



Null and alternative hypotheses:

• H₀:
• H₁:

Using Excel, you find the p-value for this test to be 0.2398. The Excel worksheets with the values of the test statistic and p-value are provided below.



Interpretation: Since the p-value (0.2398) is greater than the significance level (0.05), you fail to reject the null hypothesis. Based on the sample of investments, there is not enough evidence to conclude that the new investment strategy has significantly decreased the proportion of investments meeting or exceeding the historical target return of 50%.

watch

Check out this video on conducting a left-tailed hypothesis test to analyze investment strategies.