ame: Date: rade / Strand CENTRAL LIMIT THEOREM rections: Choose the letter that corresponds to the correct answer. Encircle the letter of your answer. 1. Which of the following is correct about the sampling distribution of the sample mean using the Central Limit Theorem? A. The mean of the sampling distribution of the means is not equal to the mean of the population. B. As the sample size n increases, the sampling distribution of the means approaches a normal distribution. C. The variance of the sampling distribution of the means is equal to the variance of the population multiplied by the sample size n. D. The standard deviation of the sampling distribution of the means is equal to the standard deviation of the population multiplied by the square root of the sample size n. 2. Which of the following statements best describes a z-value? A. The z-value leads to the area under the curve found in the normal curve table. B. The z-value leads to the area under the curve found in the normal distribution table, which is a probability. C. The z-value leads to the area under the curve found in the normal curve table, which is a percentage, and that percentage gives the desired probability for X. D. The z-value leads to the are under the curve found in the normal curve table, which is a probability, and that percentage gives the desired percentage for X. 3. Who is the French-born Mathematician who proved the first version of the Central Limit Theorem? A. Abraham de Moivre C. Aleksander Lyapunov B. Aristotle D. Rene Descartes 4. According to the Central Limit Theorem, which sample size will give a smaller standard error of the mean? A. 7 B. 12 C. 23 D. 40 5. If a population is not normally distributed, the distribution of the sample means for a given sample n will A. be positively skewed C. take the same shape as the population B. be negatively skewed D. approach a normal distribution as n increases 6. The mean and standard deviation of a population are 75 and 15, respectively. The sample size is 100. What is the standard error of the mean? A. 1.5 B. 1.73 C. 0.15 D. 8 7. The mean and standard deviation of a population are 400 and 40, respectively. Sample size is 25. What is the mean of the sampling distribution? A. 400 B. 40 C. 25 D. 8 8. The weights of the eggs produced by a certain breed of hen are normally distributed with mean 65 grams and standard deviation of 5 grams. Which of the following will you use? A. Normal Distribution C. Discrete Probability Distribution B. Central Limit Theorem D. Binomial Distribution 9. In a study done on a life expectancy of 500 people in a certain geographic region, the mean age at death was 72 years and the standard deviation was 5.3 years. If a sample of 50 people from this region is selected, and the probability that the mean life expectancy will be less than 70 years, which of the following will you use? A. Normal Distribution C. Discrete Probability Distribution B. Central Limit Theorem D. Binomial Distribution10. Which of the following refers to the standard deviation of a sampling distribution? A. It is the sum of squares. B. It is the standard error of the mean. C. It only applies only to population data. D. It can be larger than the standard deviation for the population. 1 1. What is the standard deviation of sampling distribution if standard deviation of population is 35 and sample size is 9? A. 11.67 B. 12.67 C. 13.67 D. 14.67 12. If a population has a mean of 6.5, what is the mean of the sampling distribution of its means? A. 5.6 B. 6.5 C. 6.7 D. 65 13. The mean and standard deviation of a population are 200 and 20, respectively. What is the probability of selecting 25 data values with a mean less than 190? A. 69% B. 31% C. 0.62% D. 99% For numbers 14-15, refer to the problem below. Suppose the teenagers that attend public high schools get an average of 5.7 hours of sleep each night with a standard deviation of 1.7 hours. Assume that the average sleep hour is normally distributed, and 35 high school students are randomly selected. 14. Compute the z-score for 6 hours of sleep. A. 1.04 B. 0.18 C. 0.52 D. 0.82 15. What is the probability that a randomly selected group of 35 high school students gets more than 6 hours of sleep each night? A. 0.3508 B. 0. 1492 C. 0.0714 D. 0.4286inferences about the sample and population parameters. III. ACTIVITIES A. LET US REVIEW Activity 1 Directions: Complete the table below by writing the missing information in the given table. Write your answer on your answer sheet. Conditions Illustration Area Less than z = 1 The required area is equal to 0.5 + 0.3413 = 0.8413. 1. Above z = -1 2. Between z = 0.98 and z = 2.58 3. Less than z = 1.5 B. LET US STUDY . Central Limit Theorem states that the sampling distribution of the mean approaches a normal distribution, as the sample size increases. It allows us to safely assume that the sampling distribution of the mean will be normal in most cases. . Abraham de Moivre (1667-1754), a French-born Mathematician proved the first version of the Central Limit Theorem. He used the normal distribution to approximate the distribution of the number of heads that will result when a fair coin is tossed a large number of times. Alexander Lyapunov (1857-1918) gave the first rigorous proof of the general Central Limit Theorem. . The mean of the sampling distribution of means is equal to the mean of the population. HR = M250 250 100 100 Z =-2 z = 2 z = $2 has a corresponding area of 0.4772 Since we are looking for the probability between 1150 hours and 1250 hours, the shaded part will be between -2 and 2. Add the two z-score values. P(1150