Normal distribution hypothesis

• What is the null hypothesis in normal distribution hypothesis testing?


  The null hypothesis ($H_0$) is a statement that assumes no effect or no difference in the population parameter being tested. It is the default assumption until evidence suggests otherwise.
• What is the alternative hypothesis in normal distribution hypothesis testing?


  The alternative hypothesis ($H_1$) is a statement that contradicts the null hypothesis. It represents the effect or difference we aim to detect in the population parameter.
• What is a p-value in hypothesis testing?


  The p-value is the probability of obtaining test results **at least as extreme** as the observed results, assuming the null hypothesis is true. A small p-value (typically $\leq 0.05$) suggests **rejecting the null hypothesis**.
• What is the significance level ($\alpha$) in hypothesis testing?


  The significance level ($\alpha$) is the threshold used to determine whether to reject the null hypothesis. Common values are $0.05$ or $0.01$. If the p-value is less than $\alpha$, the null hypothesis is rejected.
• What is the test statistic in normal distribution hypothesis testing?


  The test statistic is a standardized value calculated from sample data, used to determine whether to reject the null hypothesis. For a normal distribution, it is often a **z-score** or **t-score**.
• What is the critical region in hypothesis testing?


  The critical region is the range of values of the test statistic for which the null hypothesis is **rejected**. It is determined by the significance level ($\alpha$) and the type of test (one-tailed or two-tailed).
• Denoting ND hypothesis


  x̄ ~ N($\mu$,$\sigma^2$/n)
• What does the p-value represent in hypothesis testing?

  Probability of observing results under H0
• What does a p-value less than 0.01 indicate?

  Very strong evidence for rejecting H0

  We would conclude that there is highly significant evidence at the 1% level, to suggest that…… (refer to the specific context of H0 being false)
• What does a p-value between 0.001 and 0.05 suggest?

  
  

  Strong evidence for rejecting H0

  We would conclude that there is significant evidence at the 5% level, to suggest that…… (refer to the specific context of H0 being false)
• What conclusion can be drawn if p > 0.05?

  Insufficient evidence for rejecting H0

  We would conclude that at the 5% level, …… (refer to the specific context of H0 being true).
• What are the implications of different p-value thresholds in hypothesis testing?

  • P < 0.01: Very strong evidence to reject H0
  • 0.001 < p < 0.05: Strong evidence to reject H0
  • p > 0.05: Insufficient evidence to reject H0
•  Test results are normally distributed with a mean of 65 and a standard deviation of 10. After the introduction of a dynamic new teacher the results for a group of 8 students had a mean of 72. Is there evidence that the results have significantly improved at a 5% level of significance? 

   

   

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• What is the test statistic used in this context?

  The sample mean $X^{-}$
• How does the variance of the sample mean differ from the variance of the population?

  It is divided by the sample size
• What does $X^{-}$ represent?


  The mean of the sample
• What does $\mu$ represent?


  The true mean of the population
• Why do you use $X^{-}$ to make decisions?


  To make decisions about $\mu$
• What is the formula for the standardized test statistic?

  $Z =$$\frac{X^{-} - \mu}{\sigma/\sqrt{n}}$
• What does the standardized test statistic compare to?

  The critical value from the tables
• What is the critical value for a 1-tailed test at the 5% level?

  1.645
• How does the critical value relate to the standardized test statistic?

  It determines the rejection region for the hypothesis test
• Evidence


  Find test statistic using formula for Z and then compare it to the critical value from table
• Test statistic vs CV positive values

  TS>CV then reject H0

  There is sufficient evidence that...

  T.S<C.V then DONT reject H0

  There is insufficient evidence that the mean..
• Test statistic vs C.V negative values

  -T.S<-C.V then we reject H0

  There is sufficient evidence...

  -T.S>-C.V then do not reject H0

  There is insufficient evidence...