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Testing a Single Proportion
Here we will evaluate an example of hypothesis testing for a single proportion.
Learning Objective

Construct and evaluate a hypothesis test for a single proportion.
Key Points
 Our hypothesis test involves the following steps: stating the question, planning the test, stating the hypotheses, determine if we are meeting the test criteria, and computing the test statistic.
 We continue the test by: determining the critical region, sketching the test statistic and critical region, determining the pvalue, stating whether we reject or fail to reject the null hypothesis and making meaningful conclusions.
 Our example revolves around Michele, a statistics student who replicates a study conducted by Cell Phone Market Research Company in 2010 that found that 30% of households in the United States own at least three cell phones.
 Michele tests to see if the proportion of households owning at least three cell phones in her home town is higher than the national average.
 The sample data does not show sufficient evidence that the percentage of households in Michele's city that have at least three cell phones is more than 30%; therefore, we do not have strong evidence against the null hypothesis.
Term

null hypothesis
A hypothesis set up to be refuted in order to support an alternative hypothesis; presumed true until statistical evidence in the form of a hypothesis test indicates otherwise.
Full Text
Hypothesis Test for a Single Proportion
For an example of a hypothesis test for a single proportion, consider the following. Cell Phone Market Research Company conducted a national survey in 2010 and found the 30% of households in the United States owned at least three cell phones. Michele, a statistics student, decides to replicate this study where she lives. She conducts a random survey of 150 households in her town and finds that 53 own at least three cell phones. Is this strong evidence that the proportion of households in Michele's town that own at least three cell phones is more than the national percentage? Test at a 5% significance level.
1. State the question: State what we want to determine and what level of confidence is important in our decision.
We are asked to test the hypothesis that the proportion of households that own at least three cell phones is more than 30%. The parameter of interest, p, is the proportion of households that own at least three cell phones.
2. Plan: Based on the above question(s) and the answer to the following questions, decide which test you will be performing. Is the problem about numerical or categorical data? If the data is numerical is the population standard deviation known? Do you have one group or two groups?
We have univariate, categorical data. Therefore, we can perform a one proportion ztest to test this belief. Our model will be:
3. Hypotheses: State the null and alternative hypotheses in words then in symbolic form:
 Express the hypothesis to be tested in symbolic form.
 Write a symbolic expression that must be true when the original claims is false.
 The null hypothesis is the statement which includes the equality.
 The alternative hypothesis is the statement without the equality.
Null Hypothesis in words: The null hypothesis is that the true population proportion of households that own at least three cell phones is equal to 30%.
Null Hypothesis symbolically: H_{o}: p = 30%
Alternative Hypothesis in words: The alternative hypothesis is that the population proportion of households that own at least three cell phones is more than 30%.
Alternative Hypothesis symbolically: H_{o}: p > 30%
4. The criteria for the inferential test stated above: Think about the assumptions and check the conditions.
Randomization Condition: The problem tells us Michele uses a random sample.
Independence Assumption: When we know we have a random sample, it is likely that outcomes are independent. There is no reason to think how many cell phones one household owns has any bearing on the next household.
10% Condition: We will assume that the city in which Michele lives is large and that 150 households is less than 10% of all households in her community.
Success/Failure: p_{0}(n) > 10 and (1  p_{0})n > 10
To meet this condition, both the success and failure products must be larger than 10 (p_{0} is the value of the null hypothesis in decimal form. )
0.3(150) = 45 > 10 and (1 – 0.3)150 = 105 > 10
5. Compute the test statistic:
The conditions are satisfied, so we will use a hypothesis test for a single proportion to test the null hypothesis. For this calculation we need the sample proportion, p̂:
6. Determine the Critical Region(s): Based on our hypotheses are we performing a lefttailed, right tailed or twotailed test?
We will perform a righttailed test, since we are only concerned with the proportion being more than 30% of households.
7. Sketch the test statistic and critical region: Look up the probability on the table, as shown in .
Critical Region
This image shows a graph of the critical region for the test statistic in our example.
8. Determine the pvalue:
pvalue = P(z > 1.425) = 1 – P(z <1.425) = 1 0.923 = 0.077
9. State whether you reject or fail to reject the null hypothesis:
Since the probability is greater than the critical value of 5%, we will fail to reject the null hypothesis.
10. Conclusion: Interpret your result in the proper context, and relate it to the original question.
Since the probability is greater than 5%, this is not considered a rare event and the large probability tells us not to reject the null hypothesis. The pvalue tells us that there is a 7.7% chance of obtaining our sample percentage of 35.33% if the null hypothesis is true. The sample data do not show sufficient evidence that the percentage of households in Michele's city that have at least three cell phones is more than 30%. We do not have strong evidence against the null hypothesis.
Note that if evidence exists in support of rejecting the null hypothesis, the following steps are then required:
11. Calculate and display your confidence interval for the alternative hypothesis.
12. State your conclusion based on your confidence interval.
Key Term Reference
 alternative hypothesis
 Appears in these related concepts: The Null and the Alternative, Assumptions, and Example: Test for Independence
 average
 Appears in these related concepts: Outliers, Mean: The Average, and Average Value of a Function
 confidence interval
 Appears in these related concepts: Estimating a Population Proportion, What Is a Confidence Interval?, and The tDistribution
 critical value
 Appears in these related concepts: Two Regression Lines, Estimating the Target Parameter: Interval Estimation, and Estimating a Population Variance
 datum
 Appears in these related concepts: Which Average: Mean, Mode, or Median?, Averages of Qualitative and Ranked Data, and Change of Scale
 deviation
 Appears in these related concepts: Applications of Statistics, Graphs of Qualitative Data, and Variance
 hypothesis test
 Appears in these related concepts: Level of Confidence, Determining Sample Size, and Hypothesis Tests or Confidence Intervals?
 independence
 Appears in these related concepts: The Year the Polls Elected Dewey, Independence, and Departmentalization Cons
 independent
 Appears in these related concepts: Conditional Probability, Unions and Intersections, and Party Identification
 level
 Appears in these related concepts: Misleading Graphs, Randomized Design: SingleFactor, and Factorial Experiments: Two Factors
 pvalue
 Appears in these related concepts: Estimating and Making Inferences About the Slope, How Fisher Used the ChiSquared Test, and Does the Difference Prove the Point?
 population
 Appears in these related concepts: The Purpose of Statistics, Social Definition of Race, and Collecting and Measuring Data
 probability
 Appears in these related concepts: Particle in a Box, The Addition Rule, and Rules of Probability for Mendelian Inheritance
 random sample
 Appears in these related concepts: Inferential Statistics, Random Samples, and Comparing Two Independent Population Proportions
 sample
 Appears in these related concepts: Sampling, Defining the Sample and Collecting Data, and Basic Inferential Statistics
 significance level
 Appears in these related concepts: Distorting the Truth with Descriptive Statistics, Using the Model for Estimation and Prediction, and Elements of a Hypothesis Test
 standard deviation
 Appears in these related concepts: Typical Shapes, Mean, Variance, and Standard Deviation of the Binomial Distribution, and Basic Descriptive Statistics
 statistics
 Appears in these related concepts: Communicating Statistics, Understanding Statistics, and Population Demography
 ztest
 Appears in these related concepts: Alternatives to the tTest, Quantitative or Qualitative Data?, and One, Two, or More Groups?
Sources
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Cite This Source
Source: Boundless. “Testing a Single Proportion.” Boundless Statistics. Boundless, 21 Jul. 2015. Retrieved 03 Sep. 2015 from https://www.boundless.com/statistics/textbooks/boundlessstatisticstextbook/estimationandhypothesistesting12/hypothesistestingonesample54/testingasingleproportion2682719/