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In hypothesis testing, a Type I error occurs when we reject the null hypothesis even though it is true. In simpler terms, this means we think there's an effect or difference when none actually exists. Imagine this as a "false alarm."
It is often represented by the Greek letter alpha (α), which is the significance level of the test. In the context of the given exercise, the winery would make a Type I error if they decide that fewer than 25% of consumers accept screw tops, even if in reality, at least 25% do accept them.
Here’s how this error plays out:

• The winery mistakenly concludes screw tops are not acceptable to 25% of consumers.
• They might switch closure types unnecessarily, potentially impacting costs or consumer acceptance.

To control for a Type I error, we set a significance level. In this exercise, the significance level is 0.10, meaning there's a 10% risk of committing a Type I error.

A Type II error occurs when we fail to reject the null hypothesis when it's actually false. Simply put, we miss a real effect or difference, which means overlooking something important. This is like a "missed opportunity." The probability of making a Type II error is denoted by beta (β).
In the scenario given, the winery would commit a Type II error if they fail to identify that fewer than 25% of consumers actually accept screw tops when that is the truth.
Expressing it in context:

• The winery decides there is no strong evidence against the null, which means they believe at least 25% accept screw tops.
• They continue using screw tops, potentially displeasing a larger fraction of consumers who don't find them acceptable.

In this exercise, because the test statistic did not show enough evidence against the null hypothesis, the winery might have committed a Type II error, erroneously continuing the use of screw tops.

The one-sample z-test is a statistical test used to determine whether there is enough evidence to conclude that a population parameter differs from a specific value. This test is especially suitable for large sample sizes where the central limit theorem ensures a normal distribution.
In the given problem, the winery wishes to test if fewer than 25% of consumers find screw tops acceptable. Thus, a one-sample z-test for proportion is fitting.
Here’s the sequential process:

• **Calculate the sample proportion (\(\hat{p}\)):** This is done by dividing the number of consumers who accept screw tops by the total sample size.
• **Compute the standard deviation:** For a proportion, this measures how much the sample proportion can vary, calculated using the formula \(\sigma = \sqrt{\frac{p(1-p)}{n}}\) where p is the hypothesized population proportion.
• **Find the test statistic (z):** This is determined using the formula \(z = \frac{\hat{p} - p}{\sigma}\), which shows how far the sample proportion is from the hypothesized proportion, in standard deviation units.
• **Decision rule:** Compare the test statistic to a critical value from the z-table that corresponds to the significance level. A significant outcome leads to rejecting the null hypothesis.

In this exercise, the calculated z-value did not reach the critical threshold, so the evidence wasn't strong enough to support the alternative hypothesis. This directs the winery to retain their null proposition. Employing a one-sample z-test ensures that the winery bases its decision on statistical evidence rather than mere assumptions.