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Data analysis: hypothesis testing
Data analysis: hypothesis testing

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7.2 Test your understanding

In this section you will try a problem on your own. Using the following activity, you can test your understanding of how to calculate the z-score for a problem related to the population and the sample proportion.

Activity 9 Testing proportion hypothesis

Timing: Allow approximately 45 minutes to complete this activity

According to records, 70% of customers are willing to spend £15 on a 3D movie ticket. A ticket agent believes that this value is different. He surveys 200 individuals and found that 128 stated their willingness to spend a different amount on a ticket to a 3D movie. At the 95% confidence level, is there enough evidence to reject the null hypothesis?

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Discussion

Step 1: State the hypotheses.

The first step to tackle all the problems concerning hypothesis tests is to state the null and alternative hypotheses.

H0: 70% of customers are willing to spend £15 on a ticket to a 3D movie.

Ha: 70% of customers are not willing to spend £15 on a ticket to a 3D movie.

Or it can be written as

H0: p = 70% (or 0.7)

Ha: p ± 70%

Step 2: Use a z-score formula to calculate the z-score.

cap z equals p hat minus p sub zero divided by square root of p sub zero times left parenthesis one minus p sub zero right parenthesis divided by n

p hat = sample (observed) proportion = 128 divided by 200 = 0.64

p0 = population proportion = 0.70

n = sample size = 200

Using these values, you can calculate the z-score.

equation sequence part 1 cap z equals part 2 0.64 minus 0.70 divided by Square root of 0.70 times left parenthesis one minus 0.70 right parenthesis divided by 200 equals part 3 negative 1.85

Step 3: Determine p-value using the Excel formula ‘NORM.S.DIST(z, cumulative)’

Judging from the null and alternative hypotheses, this is a two-tailed test.

Therefore, you need to divide the levels of significance by 2.

At 95% confidence level, the significance level equals 1 − 95%. Thus, α = 0.05

0.05 divided by two equals 0.025.

In order to reject the null hypothesis at 95% confidence levels in a two-tailed test, the p-value needs to be less than 0.025.

Given the z-score is negative, you can use Table  5 to determine the p-value. You can find the area left of z corresponding to a z-score of 1.85 is 0.0322.

Step 4: Make a decision based on the p-value.

The p-value from the calculation = 0.0322, which is ≥ 0.025.

Thus, you will not reject the null hypothesis that claims 70% of customers are willing to spend £15 on a ticket to a 3D movie.

Just a quick note: if this is not a two-tailed test (i.e. if it is a one tailed-test, associated with the left tail), you will not divide the significance levels by 2. In this situation, you will still use α = 0.05. So, as 0.0322 (p-value) 0.

This section aims to demonstrate how to use the z-score to make inferences about the population proportion based on sample data. When the population proportion is known, you can calculate the z-score to determine the likelihood of observing a particular sample proportion.