Problem
MATH1127: Introduction to Statistics (60022) ... Lesson 9.3 Distribution Needed for Hypothesis Testing score: $4 L / 100 \quad 6 / 14$ answered Question 12 Testing: \[ \begin{array}{r} H_{0}: \mu \geq 13.9 \\ H_{1}: \mu<13.9 \end{array} \] Your sample consists of 20 values, with a sample mean of 13.6. Suppose the population standard deviation is known to be 1.5 . a) Calculate the value of the test statistic, rounded to 2 decimal places. \[ z= \] b) At $\alpha=0.02$, the rejection region is \[ \begin{array}{l} z>2.05 \\ z>2.33 \\ z<-2.05 \\ z<-2.33 \\ z<-2.05 \text { or } z>2.05 \\ z<-2.33 \text { or } z>2.33 \end{array} \] c) The decision is to Accept the null hypothesis Accept the alternative hypotheis IIM 2842.jpg IMG 2841.jpg MG $2840 . j p g$
Solution
Step 1 ：Given that the population mean (mu) is 13.9, the sample mean (x_bar) is 13.6, the population standard deviation (sigma) is 1.5, and the sample size (n) is 20.
Step 2 ：We calculate the test statistic (z) using the formula: \(z = \frac{x_{bar} - \mu}{\frac{\sigma}{\sqrt{n}}}\)
Step 3 ：Substituting the given values into the formula, we get: \(z = \frac{13.6 - 13.9}{\frac{1.5}{\sqrt{20}}}\)
Step 4 ：Solving the above expression, we find that the value of the test statistic, rounded to 2 decimal places, is \(\boxed{-0.89}\)
