1. **Problem 29:** Find the t-value for df = 11 where the area to the left is 0.025. - This corresponds to the lower 2.5% tail of the t-distribution with 11 degrees of freedom. - From the t-table, the critical value for 0.025 in the left tail with df=11 is approximately $-2.201$. **Answer:** C) -2.201 2. **Problem 30:** Which factor does NOT affect the width of a confidence interval? - Width depends on sample size, population variance (or standard deviation), and confidence level. - Sample mean does NOT affect the width; it affects the center. **Answer:** A) Sample mean 3. **Problem 31:** Find $A$ such that $P(|t| \geq A) = 0.01$ with sample size 25 (df=24). - This is a two-tailed test with total tail area 0.01, so each tail has 0.005. - From t-table for df=24 and 0.005 in one tail, $A \approx 2.797$ (closest given is 2.756). **Answer:** C) 2.756 4. **Problem 32:** For 80% confidence interval with known variance $\sigma=40$, which critical value to use? - Known variance means use z-distribution. - For 80% confidence, critical z-value is $\pm 1.28$. **Answer:** B) ± 1.28 5. **Problem 33:** Interpretation of 99.7% confidence interval (12.64 cm, 14.44 cm). - Correct interpretation: If we repeat sampling many times, 99.7% of such intervals contain the true mean. **Answer:** C) If we were to repeat this sampling many times, 99.7% of the confidence intervals we could construct would contain the true population mean. 6. **Problem 34:** Critical z-value for 85% confidence interval. - For 85%, tail area is 0.075 each side. - From z-table, critical z ≈ 1.44. **Answer:** B) 1.44 7. **Problem 35:** Critical values for two-tailed test with $\alpha=0.01$ using z-table. - Each tail has 0.005. - Critical z ≈ ±2.58. **Answer:** E) ± 2.58 8. **Problem 36:** p-value = 0.25 for alternative hypothesis that less than 10% are left-handed. - p-value > typical alpha (0.05), so fail to reject null. - Cannot conclude more than 10% are left-handed. **Answer:** D) We cannot conclude that more than 10% of the population is left-handed. 9. **Problem 37:** Critical value at $\alpha=0.05$ for sample mean test with n=49. - Large sample, use z critical value for one-tailed test at 0.05: z = -1.645 (left tail). **Answer:** C) z = -1.645 10. **Problem 38:** Rejection probability of null hypothesis when true is called? - This is the Type I error rate, also called level of significance. **Answer:** B) Level of Significance 11. **Problem 39:** Hypothesis tested for rejection assuming it is true is the? **Answer:** A) Null Hypothesis 12. **Problem 40:** Type II error is committed when? - Type II error: fail to reject null when it is false. **Answer:** C) We accept a null hypothesis when it is not true. 13. **Problem 41:** Table value for 90% confidence interval with sample size 20, unknown population SD. - df = 19, t critical ≈ 2.064. **Answer:** B) t = 2.064 14. **Problem 42:** Calculate standard error for sample mean with n=100, SD=40. - Standard error $= \frac{40}{\sqrt{100}} = 4.00$. **Answer:** A) 4.00 15. **Problem 43:** Appropriate hypothesis for mean unpaid balance at most 250, sample mean 275. - Null: $\mu \leq 250$, Alternative: $\mu > 250$. **Answer:** B) Ho: μ ≤ 250 H1: μ > 250 16. **Problem 44:** Critical value at $\alpha=0.05$ for test in problem 43. - One-tailed test, z critical = 1.645. **Answer:** A) z = 1.645 17. **Problem 45:** Test statistic for sample mean 275, SD 25, n=49, testing mean ≤ 250. - $Z = \frac{275 - 250}{25/\sqrt{49}} = \frac{25}{25/7} = 7$. **Answer:** E) Z_calc = 7 18. **Problem 46:** Hypothesis to test lecturer's claim mean = 60, sample mean 63, SD 6, n=25. - Two-tailed test: Ho: μ=60, H1: μ ≠ 60. **Answer:** A) Ho: μ = 60 H1: μ ≠ 60 19. **Problem 47:** Which statement is NOT true about hypothesis tests? - Hypotheses are about populations, not samples. **Answer:** C) Hypotheses are statements about the sample (or samples) from the population. 20. **Problem 48:** Strongest evidence to reject $H_0: \mu \geq 30$ with sample size 36. - Smaller sample mean and smaller SD give stronger evidence. - Sample mean 25 with SD 6 is strongest. **Answer:** E) sample mean = 25, sample standard deviation = 6 21. **Problem 49:** If hypothesis not rejected at 5% level, then? - It will not be rejected at 10% level (less strict). **Answer:** E) will not be rejected at the 10% level 22. **Problem 50:** Reject $H_0$ if standardized test statistic exceeds critical values at $\alpha=0.01$, n=32. - For two-tailed test, critical z = ±2.575. **Answer:** A) Rejected Ho if the standardised test statistic is greater than 2.575 or less than -2.575