2.5.3 Hypothesis testing for the normal distribution

• What does the null hypothesis state in hypothesis testing?

  No significant difference
• The alternative hypothesis contradicts the null hypothesis.
• What type of error is committed when rejecting a true null hypothesis?

  Type I error
• Common values for the significance level include 0.05, 0.01, and 0.10
• Lower significance levels are used when falsely rejecting the null hypothesis has serious repercussions.
• Match the significance level with its Type I error probability:

  0.05 (5%) ↔️ 5%

  0.01 (1%) ↔️ 1%

  0.10 (10%) ↔️ 10%
• What is the definition of a Type I error in hypothesis testing?

  Rejecting a true null hypothesis
• The choice of significance level depends on the consequences of a Type I error
• What is the formula for calculating the test statistic in a normal distribution hypothesis test?

  z = \frac{\bar{x} - μ}{\frac{σ}{\sqrt{n}}}</latex>
• In the test statistic formula, $σ$ represents the population standard deviation.
• What is the first step in determining the critical region in hypothesis testing?

  Identify the significance level
• The critical region is the set of values for the test statistic that lead to rejecting the null hypothesis
• In hypothesis testing, $μ$ represents the population mean stated in the null hypothesis.
• $σ$ is the sample standard deviation.


  False
• In an example, we have a sample of 36 apples with a mean weight of 105g</latex>. The null hypothesis states that the population mean weight is $100g$. The population standard deviation is $10g$. Using the formula, the calculated z-value is approximately 3.00.
• Steps to determine the critical region in hypothesis testing

  1️⃣ Identify the significance level $(α)$.

  2️⃣ Determine if the test is one-tailed or two-tailed.

  3️⃣ Use standard normal distribution tables to find the critical value(s) corresponding to $(α)$.
• Match the significance level with its critical value for a one-tailed test:

  0.05 ↔️ 1.645

  0.01 ↔️ 2.33

  0.10 ↔️ 1.28
• The null hypothesis states that there is a significant difference or effect.

  False
• A Type I error occurs when we reject a true null hypothesis.
• What is the formula for calculating the test statistic in a normal distribution?

  z = \frac{\bar{x} - μ}{\frac{σ}{\sqrt{n}}}</latex>
• If the calculated z-value is 3.00, it is greater than the critical value for a one-tailed test with $α =$$0.05$.
• For a one-tailed test with $α =$$0.05$, the critical value is 1.645.
• In hypothesis testing, we can reject the null hypothesis if the test statistic falls within the critical region.
• What should you use to find critical values for hypothesis testing?

  Standard normal distribution tables
• The critical value for a one-tailed test with $α =$$0.05$ is 1.645
• The critical value for a two-tailed test with $α =$$0.01$ is ±2.58.
• What is the critical value for a one-tailed test with $α =$$0.10$?


  1.28
• Steps to make a decision in hypothesis testing

  1️⃣ Compare the test statistic to the critical value

  2️⃣ Compare the p-value to α

  3️⃣ Reject H0 if the test statistic is more extreme or the p-value is less than α

  4️⃣ Fail to reject H0 otherwise
• What should you do if the test statistic is more extreme than the critical value?

  Reject H0
• If the p-value is greater than α, you should reject H0.

  False
• For a p-value less than α, you should reject H0.
• What does it mean if you reject H0?

  Significant evidence supports H1
• If you fail to reject H0, it means there is enough evidence to support H1.

  False