7.4 Setting Up a Test for a Population Mean

• To set up a test for a population mean, it is crucial to identify the research question and formulate the null and alternative hypotheses
• The significance level ($\alpha$) in hypothesis testing determines the threshold for rejecting the null hypothesis
• Choosing $\alpha$ depends on the consequences of Type I errors versus the need for sensitivity


  True
• A z-test is used when the population standard deviation is known


  True
• What must be determined about the population standard deviation before choosing a test statistic?

  Known or unknown
• What does the null hypothesis in a hypothesis test assert about the population mean?

  No difference from a value
• What is the common value for the significance level ($\alpha$) in general use?


  0.05
• What determines the choice between a z-test and a t-test for a population mean?

  Known or unknown standard deviation
• Match the test statistic with the condition for its use:

  z-test ↔️ Population standard deviation known

  t-test ↔️ Population standard deviation unknown
• When should a t-test be used instead of a z-test?

  Population SD is unknown
• What is the formula for the z-test statistic?

  z = \frac{\bar{x} - \mu_{0}}{\frac{\sigma}{\sqrt{n}}}</latex>
• What is the formula for the t-test statistic?

  t = \frac{\bar{x} - \mu_{0}}{\frac{s}{\sqrt{n}}}</latex>
• In the example calculation, the t-test statistic is 2.5
• The p-value is compared to the significance level to make a decision
• The null hypothesis asserts there is no difference between the population mean and a specified value


  True
• If $\alpha =$$0.05$, there is a 5% chance of rejecting the null hypothesis when it is true


  True
• Match the assumption/condition with its description:

  Independence ↔️ Sample observations must be independent of each other

  Normality ↔️ The population from which the sample is drawn must be normally distributed

  Known Population Standard Deviation ↔️ If unknown, the sample size must be large enough
• A t-test is used when the population standard deviation is unknown
• If the population standard deviation is known, a z-test should be used.
• The alternative hypothesis for a two-tailed test is denoted as \( H_a: \mu \neq \mu_0 \), where \neq means "not equal to."
• Steps to check assumptions and conditions for a hypothesis test

  1️⃣ Independence of observations

  2️⃣ Normality of the population

  3️⃣ Known population standard deviation (or large sample size if unknown)
• A z-test is used when the population standard deviation is known because the test statistic follows a standard normal distribution.
• The formula for the z-test statistic is \( z = \frac{\bar{x} - \mu_{0}}{\frac{\sigma}{\sqrt{n}}} \), where \sigma is the population standard deviation.
• The t-test is more appropriate when the population standard deviation is unknown
• In the z-test formula, $\bar{x}$ represents the sample mean
• In the t-test formula, $s$ represents the sample standard deviation
• What does the p-value represent?

  Probability of extreme test statistic
• What decision is made if the p-value is less than or equal to the significance level?

  Reject null hypothesis
• The p-value assumes the null hypothesis is true.


  True
• Rejecting the null hypothesis means the population mean is likely different from the specified value.


  True
• Match the alternative hypothesis with its notation:

  States the population mean is different from the specified value ↔️ $H_{a}: \mu \neq \mu_{0}$

  States the population mean is greater than the specified value ↔️ $H_{a}: \mu > \mu_{0}$

  States the population mean is less than the specified value ↔️ $H_{a}: \mu < \mu_{0}$
• The significance level ($\alpha$) determines the threshold for rejecting the null hypothesis and is the probability of making a Type I error
• The choice between a z-test or a t-test depends on whether the population standard deviation is known or unknown
• The t-test accounts for additional uncertainty in estimating the standard deviation from the sample

  True
• The t-test is more appropriate when the population standard deviation is unknown because it accounts for additional uncertainty.


  True
• The significance level ($\alpha$) represents the probability of rejecting a true null hypothesis.


  True
• The population from which the sample is drawn must be normally distributed for a valid hypothesis test.
• The t-test is appropriate for small sample sizes when the population standard deviation is unknown.


  True
• What is the key difference in the t-test formula compared to the z-test formula?

  Use of sample standard deviation
• The z-test is used when the population standard deviation is known.


  True