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The null hypothesis is a fundamental concept in hypothesis testing, serving as an initial assumption or claim about a population parameter, such as a proportion or mean. In our case study, the null hypothesis (\(H_0: p = 0.80\)) implies that \(80\%\) of all supermarket prices end in the digits 9 or 5. This hypothesis stands as our default position, and we need substantial evidence to reject it.

When we conduct a hypothesis test, the aim is to determine if there is enough statistical evidence to support an alternative hypothesis, suggesting a different scenario.

• The null hypothesis always includes an equality (e.g., \(p = 0.80\)).
• The alternative hypothesis in our test aims to show that the observed sample proportion is less than 80%.

Hypothesis testing typically involves comparing the observed data against what the null hypothesis predicts, allowing us to decide whether to reject the null hypothesis or not.

Sample proportion is a key concept when making inferences about a population proportion. It is calculated from the sample data and used to estimate the parameter of interest.
In the supermarket pricing example, we took a random sample of 115 items to examine how many prices ended in the digits 9 or 5, which helps us form a sample proportion.

To calculate the sample proportion (\(\hat{p}\)), we divided the number of items meeting our criterion (88) by the total number of items in our sample (115): \(\hat{p} = \frac{88}{115} \approx 0.765\).

• The sample proportion is our best estimate of the true population proportion.
• It serves as the basis for conducting hypothesis tests and calculating confidence intervals.

The reliability of the sample proportion is influenced by the sample size; thus, larger samples generally result in more accurate estimates.

The Z-test for proportions is a statistical method used to test hypotheses about a population proportion. It is particularly useful when you have a sample proportion and want to compare it with a known population proportion, as seen in this exercise.
The process involves several steps: calculating the standard error, finding the test statistic, and comparing it against a critical value.

In our example:

• The standard error (SE) measures the variability of the sample proportion and is calculated via the formula: \(SE = \sqrt{\frac{p(1-p)}{n}}\).
• We used the hypothesized proportion (\(p = 0.80\)), obtaining a SE of approximately 0.037.
• The Z-test statistic is computed using the formula: \(z = \frac{\hat{p} - p}{SE}\).

For our test, the calculated Z-value was approximately -0.946. This is then compared to a critical value for the chosen significance level (\(\alpha = 0.05\)), which was about -1.645 for a one-tailed test.
Since the calculated Z-value did not exceed the critical value, we did not find strong evidence to reject the null hypothesis, concluding that the true proportion of prices ending in 9 or 5 is still possibly 80% or more.