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In hypothesis testing, the null hypothesis is a statement about the general population or a specific parameter of interest that we assume to be true until evidence suggests otherwise. It's essentially the status quo or default position that there is no change or effect. In our supermarket pricing exercise, the null hypothesis, denoted as \( H_0 \), posits that the actual proportion \( p \) of supermarket prices ending in the digits 9 or 5 is 80%. This hypothesis can be expressed as:

• \( H_0: p = 0.80 \)

It's important to understand that the null hypothesis does not suggest what we personally think is true. Instead, it stands as a benchmark against which we test our sample data. If the sample evidence strongly contradicts it, we may consider the alternative hypothesis. Here, the alternative hypothesis is that less than 80% of prices end in a 9 or 5:

• \( H_a: p < 0.80 \)

The sample proportion is a key concept in statistics that represents the fraction of observations in a sample that satisfy a specified condition. It is calculated by dividing the number of successes by the total number of observations. In this exercise, the 'success' is defined as a price ending in the digits 9 or 5. Given a random sample of 115 items, where 88 prices meet this condition, the sample proportion \( \hat{p} \) is calculated as follows:

• \( \hat{p} = \frac{88}{115} \approx 0.7652 \)

This sample proportion provides an estimate of the true population proportion and is key in comparing against the hypothesized value under the null hypothesis. When a sample proportion is further from the null hypothesis value, it suggests greater evidence against the null. In this case, the sample proportion of approximately 76.52% is slightly lower than 80%, prompting further analysis in hypothesis testing.

The test statistic in a hypothesis test serves as an important measure that tells us how far our sample proportion is from the hypothesized population proportion stated in the null hypothesis. For proportion tests, the test statistic follows a standard normal distribution, allowing us to calculate the \( z \)-score, which quantifies the distance in standard deviations of the sample proportion from the null hypothesis proportion. The formula for calculating the \( z \)-score is:\[ z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}} \]Where:

• \( \hat{p} \) is the sample proportion.
• \( p_0 \) is the null hypothesis proportion.
• \( n \) is the sample size.

For the exercise, by substituting \( \hat{p} = 0.7652 \), \( p_0 = 0.80 \), and \( n = 115 \), the test statistic \( z \) is computed as approximately -0.935. This negative value indicates that the sample proportion is below the hypothesized value, but we must compare it with a critical value to determine significance.

The critical value acts like a threshold in hypothesis testing, determining whether the test statistic lies in the critical region, thus leading to the rejection of the null hypothesis. For a single-tailed test like the one in this exercise, where we're checking if the proportion is less than a specified value, the critical value corresponds to the chosen significance level \( \alpha \), here 0.05.Using a standard normal (or \( z \)-) distribution table, the critical value for a left-tailed test with \( \alpha = 0.05 \) is approximately -1.645. This value implies the cutoff point at which the probability of observing a test statistic as extreme as, or more extreme than, the critical value under the null hypothesis is less than 5%.For our test statistic of -0.935, because it's greater than -1.645, it falls outside the critical region. Therefore, we do not reject the null hypothesis, indicating insufficient evidence to suggest that less than 80% of supermarket prices end in 9 or 5.