QUESTION 13 An online stock trading company makes part of their revenue from clients when the clients trade stocks therefore, it is important to the company to have an good idea of how many trades its clients are making in a given year. In a sample of 120 clients of an online stock trading company, the average number of trades per year was 82 with a standard deviation of 16 . If you were to test the hypothesis that the average number of trades per year is different than the previous year when the average number of trades was 85 (using the \( 5 \% \) level of significance), what is p-value for this test? (please round your answer to 4 decimal places)

To calculate the p-value for the hypothesis test, you can follow these steps: 1. **Hypotheses formulation**: - Null hypothesis (\( H_0 \)): \( \mu = 85 \) (the average number of trades is the same as the previous year) - Alternative hypothesis (\( H_a \)): \( \mu \neq 85 \) (the average number of trades is different from the previous year) 2. **Calculate the test statistic** using the formula for a t-test: \[ t = \frac{\bar{x} - \mu_0}{\frac{s}{\sqrt{n}}} \] Where: - \( \bar{x} = 82 \) (sample mean) - \( \mu_0 = 85 \) (population mean under the null hypothesis) - \( s = 16 \) (sample standard deviation) - \( n = 120 \) (sample size) 3. **Plug in the values**: \[ t = \frac{82 - 85}{\frac{16}{\sqrt{120}}} = \frac{-3}{\frac{16}{10.954}} \approx \frac{-3}{1.461} \approx -2.051 \] 4. **Find the degrees of freedom (df)**: \[ df = n - 1 = 120 - 1 = 119 \] 5. **Calculate the p-value** for a two-tailed test using a t-distribution: You can look up the t value of approximately -2.051 in the t-table for \( df = 119 \), or you can use a calculator/statistical software. The approximate p-value for \( t = -2.051 \) for \( df = 119 \) is about 0.0420. Thus, the p-value for this test is **0.0420** (rounded to four decimal places).