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In hypothesis testing, the significance level is a critical concept that determines the threshold at which you decide whether a sample result is statistically significant. It is denoted by the Greek letter \( \alpha \) and usually set at values such as 0.05, 0.01, or 0.10. The significance level indicates the probability of rejecting the null hypothesis when it is actually true.

In the context of our exercise, we used a significance level of 0.05. This means we were willing to accept a 5% chance of incorrectly rejecting the null hypothesis, which claims the error rate of scanned prices is 2%.

- A lower significance level often means you're being more conservative and less likely to make a Type I error (false positive).- Hypothesis tests can be one-tailed (checking for an effect in one direction) or two-tailed (checking both directions). Here, we conducted a right-tailed test, hypothesizing the error rate is greater than 2%, aligning with the direction of our alternative hypothesis.

Normal approximation is a method used to simplify calculations in statistics, especially for hypothesis testing. When dealing with a large sample size, like our 200 purchase records, it is common to approximate the binomial distribution with a normal distribution.

This approximation is based on two key conditions:

• The number of trials (\( n \)) is large enough.
• The expected number of successes and failures (\( np \) and \( n(1-p) \)) are both greater than or equal to 5.

In our exercise, since we are dealing with a proportion (\( ychside \)), we approximated the test statistic using the normal distribution’s Z-score formula:\(Z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}\).
This calculation provided a way to compare observed results to what is generally expected, determining significance.

A Type II error occurs in hypothesis testing when we fail to reject the null hypothesis, even though the alternative hypothesis is true. This is often denoted as \( \beta \) and demonstrates the test’s power, representing the probability of missing a real effect.

In our example, we calculated the probability of making a Type II error given that the true mistake rate is 5% and the tested rate is still assumed to be 2%. We adjusted the formula for our test statistic, recalculating with the new assumed rate:
\(Z = \frac{0.05 - 0.02}{\sqrt{\frac{0.02\times0.98}{200}}}\),and explored the area under the normal curve up to the critical value from this Z-score.

- A high Type II error probability suggests low test power, indicating the test might not be sensitive enough to detect effects if they exist in the accessed population.- Reducing Type II errors often involves increasing the sample size or selecting a significance level that effectively balances Type I and Type II risks.