7.6 Concluding a Test for a Population Mean

• The null hypothesis states that the population mean is equal to a specified value, denoted as μ₀
• Match the statistical variable with its symbol:

  Sample Mean ↔️ $\bar{x}$

  Population Mean ↔️ $\mu$

  Sample Standard Deviation ↔️ $s$

  Sample Size ↔️ $n$
• The degrees of freedom for a t-test are typically calculated as n-1
• The null hypothesis for a population mean assumes that the population mean is equal to a specific value
• If the population is not normal, a large sample size (n ≥ 30) allows the use of the Central Limit Theorem
• The formula for calculating the t-statistic is $t =$$\frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}}}$
• The t-statistic standardizes the difference between the sample mean and population mean to determine if the null hypothesis should be rejected.

  True
• The p-value represents the probability of obtaining a test statistic at least as extreme as the one observed, assuming the null hypothesis is true.


  True
• What decision do you make if the p-value is less than the significance level?

  Reject the null hypothesis
• Rejecting the null hypothesis means the population mean is equal to the specified value.

  False
• What is the symbolic representation of the alternative hypothesis for a population mean?

  H₁: μ ≠ μ₀
• Observations within the sample must be independent if the sample size is no more than 10% of the population size

  True
• Steps to determine the p-value after calculating the t-statistic

  1️⃣ Identify the degrees of freedom (df)

  2️⃣ Use a t-distribution table or software

  3️⃣ Find the p-value associated with t and df

  4️⃣ The p-value represents the probability of obtaining a test statistic at least as extreme as observed
• The significance level is typically 0.05 or 0.01 in hypothesis testing.


  True
• Match the hypothesis statement with its symbol:

  Null Hypothesis ↔️ $H₀: \mu =$$\mu₀$

  Alternative Hypothesis ↔️ $H₁: \mu ≠ \mu₀$
• Meeting the conditions for inference ensures the reliability and accuracy of the test results.


  True
• The population mean in the t-statistic formula is the mean under the null hypothesis
• To find the p-value, one can use a t-distribution table or a calculator/software with the calculated t-statistic and the degrees of freedom
• The validity of the p-value depends on meeting the conditions for inference, such as random, normal, and independent data.

  True
• What decision do you make if the p-value is 0.03 and the significance level is 0.05?

  Reject the null hypothesis
• What is the symbolic representation of the null hypothesis for a population mean?

  H₀: μ = μ₀
• If the population is not normal, the sample size must be at least 30
• The t-statistic formula for testing a population mean is standardizes the difference between the sample mean and population mean.
• What is the p-value in hypothesis testing?

  Probability of extreme results
• What does the alternative hypothesis assert in a hypothesis test for a population mean?

  Population mean is not equal
• What does the independence condition require in hypothesis testing?

  Sample size ≤ 10% of population
• What does the variable \(\bar{x}\) represent in the t-statistic formula?

  Sample mean
• What is the formula to calculate degrees of freedom (df) in a t-test?

  df = n - 1
• What are the typical significance levels used in hypothesis testing?

  0.05 or 0.01
• If the calculated p-value is greater than or equal to the significance level, you fail to reject the null hypothesis
• The null hypothesis (H₀) assumes the population mean is equal to a specified value
• Match the hypothesis with its symbolic representation:

  Null Hypothesis ↔️ H₀: μ = μ₀

  Alternative Hypothesis ↔️ H₁: μ ≠ μ₀