1. **Problem Statement:** We have the claim that the average age of year two students at MUBS is 23 years. We want to test this claim using hypothesis testing, based on samples from the student population. 2. **(a) What is hypothesis testing?** Hypothesis testing is a statistical method used to make decisions about a population parameter based on sample data. It helps to determine whether there is enough evidence to reject a stated assumption (the null hypothesis). 3. **Procedure for conducting hypothesis testing:** 1. Formulate the null hypothesis ($H_0$) and alternative hypothesis ($H_a$). 2. Choose the significance level ($\alpha$), commonly 0.05 (5%). 3. Select the appropriate test statistic based on the sample data and distribution. 4. Calculate the test statistic from the sample. 5. Determine the critical value(s) or p-value corresponding to the test statistic. 6. Make a decision: reject $H_0$ if test statistic falls in critical region or p-value < $\alpha$; otherwise, fail to reject $H_0$. 4. **(b)(i) Hypothesis testing for average age with small and large samples:** **Given:** - Population mean claim $\mu_0 = 23$ - Sample size $n=81$ (for large sample) - Sample mean $\bar{x} = 22$ - Sample variance $s^2=1$ meaning standard deviation $s=1$ - Significance level $\alpha=0.05$ **Hypotheses:** - $H_0: \mu = 23$ (Null hypothesis: average age is 23) - $H_a: \mu \neq 23$ (Alternative: average age is not 23) **Large sample test (using z-test approximation):** $$ z = \frac{\bar{x} - \mu_0}{\frac{s}{\sqrt{n}}} = \frac{22 - 23}{\frac{1}{\sqrt{81}}} = \frac{-1}{\frac{1}{9}} = -9 $$ Critical z-values for two-tailed test at $\alpha=0.05$ are $\pm 1.96$. Since $z=-9$ is less than $-1.96$, we reject $H_0$. There is strong evidence the average age is not 23. **Small sample test:** If a smaller sample is used (say $n<30$), we use the t-distribution: The test statistic is: $$ t = \frac{\bar{x} - \mu_0}{\frac{s}{\sqrt{n}}} $$ The critical value depends on degrees of freedom $df = n-1$ and $\alpha=0.05$. Since the problem specifies 81 students (large), actual small sample data is not given, but the test would follow above using t-distribution. 5. **(b)(ii) Point estimate vs interval estimate:** - A **point estimate** is a single value estimate of a parameter, e.g., sample mean $\bar{x}$ estimating population mean $\mu$. - An **interval estimate** provides a range of values (confidence interval) within which the parameter is expected to lie with a certain probability, e.g., a 95% confidence interval. 6. **(b)(iii) Differences in conclusions using small vs large samples:** - Larger samples provide more reliable estimates and the z-test can be used. - Smaller samples have more variability and require the t-test which accounts for additional uncertainty. - The conclusion may differ because small samples have wider confidence intervals and less power to detect differences. In this case, the large sample test clearly rejects $H_0$; small sample results depend on sample data but generally are less conclusive. Final answers: - Large sample: reject null hypothesis. - Small sample: depends on sample size and t-test results.