Statistics Probability Quarter 3 Module 6: Central Limit Theorem PDF

Preview

Summary

Statistics & Probability – Grade 11 Quarter 3 – Module 6: Central Limit TheoremWhat I KnowDirections: Read and analyze the story below. Write the letter of the correct answer on your answer sheet. Which sample size gives a smaller standard error of the mean according to Central Limit theorem? A....

Description

Statistics & Probability – Grade 11 Quarter 3 – Module 6: Central Limit Theorem What I Know Directions: Read and analyze the story below. Write the letter of the correct answer on your answer sheet. 1. Which sample size gives a smaller standard error of the mean according to Central Limit theorem? A. 10 B. 15 C. 25 D. 35 2. The Central Limit Theorem says that the sampling distribution of the sample mean is approximately normal if __________. A. the sample size is large. B. all possible sample are selected. C. the standard error of the sampling mean is small. D. none of the above. 3. The mean of the sampling distribution of the sample means according to the Central Limit Theorem is __________. A. exactly equal to the population mean. B. close to the population mean if the sample size is large. C. equal to the population mean divided by the square of the sample size. D. cannot be determined. 4. The standard deviation of the sampling distribution of the sample means according to the Central Limit Theorem is __________. A. exactly equal to the standard deviation. B. close to the population standard deviation if the sample size is large. C. equal to the population standard deviation divided by the square root of the sample size. D. cannot be determined

5. Samples of size 25 are selected from a population with a mean of 40 and a standard deviation of 7.5. What is the mean of the sampling distribution of sample means? A. 7.5 B. 8 C. 25 D. 40 6. Samples of size 25 are selected from a population with a mean of 40 and a standard deviation of 7.5. What is the standard error of the sampling distribution of sample means? A. 0.3 B. 1.5 C. 7.5 D. 8 7. What happens to the shape of a sampling distribution of sample means as n increases? A. It becomes narrower and more normal C. It becomes narrower and bimodal B. It becomes wider and more normal D. It becomes wider and skewed right 8. In a group of 20 randomly selected unicorns, the mean is 1,000 and the standard deviation is 25, what is the standard deviation of the sampling distribution? A. 25 B. 5.59 C. 4 D. 1.25 9. A group of 625 students has a mean age of 15.8 years with a standard deviation of 0.6 years. The ages are normally distributed. What is the probability that a randomly selected students are older than 16.5 years old? A. 12.1% B. 86.4% C. 87.9% D. 13.6% 10. If the random samples are large, what is the shape of the sampling distribution of the mean? A. skewed to the left C. normal B. skewed to the right D. rectangular 11. These symbols  and  represent the mean and standard deviation for which of the following choices?

A. The Population Distribution B. The Sample

C.

The

Sampling

D. None of these

For numbers 12-15, refer to the problem below. The average precipitation for the first 7 months of the year is 19.32 inches with a standard deviation of 2.4 inches. Assume that the average precipitation is normally distributed. 12. What is the average precipitation of 5 randomly selected years for the first 7 months? A. 19.32 in B. 22.4 in C. 20.52 in D. 15.56 in 13. What is the probability that a randomly selected year will have precipitation greater than 18 inches for the first 7 months? A. 0.7088 B. 0.8523 C. 0.4562 D. 0.1258 14. Compute the z-score for precipitation of 18 inches for 5 randomly selected years for the first 7 months. A. – 0.55 B. 0.55 C. 1. 23 D. – 1.23 15. What is the probability of 5 randomly selected years will have an average precipitation greater than 18 inches for the first 7 months? A. 0.8907 B. 0.2587 C. 0.4156 D. 0.2879

Lecture 1: Central Limit Theorem Statistics

is

the

most

commonly

used

branch

of

mathematics. We use it almost every day. It is also a must-have

knowledge for a data scientist. Central Limit Theorem is the cornerstone of it. In statistics, the given data set represents a sample from the entire population. Using this sample, we try to see the patterns in the data. We then try to generalize the patterns in the sample to the population while making the predictions. Central limit theorem helps us to make inferences about the sample and population parameters. What’s In The context of Central Limit Theorem is comprised of statistical terms. To test your prior knowledge of the concepts, let’s see if you have understood the terms listed by answering the activity on the next page.

Match the definitions in Column A with the corresponding statistical terms in Column B. Write the letter of the correct answer on your answer sheet. Column

Column A

B

A. Mean B. Population C. Sample D. Distribution E. Standard Deviation

F.

Sample Size

G. Normal Distribution

1. The set that contains all data of elements, individuals or measurements from your

experimenting space.

2. It describes the data/population /sample range and how data is spread in that range. 3. Average value of all data from your population or sample. 4. It is a randomly selected subset from

the

population where the sample size is denoted by n.

5. It describes how spread the population is. 6. The population is spread perfectly symmetrical with (σ) standard deviations around the mean value. What’s New Central Limit Theorem states that the sampling distribution of the mean approaches a normal distribution, as the sample size increases. Regardless of the initial shape of the population distribution, if samples of size n are randomly selected from a population, the sampling distribution of the sampling means will approach a normal distribution as the sample size n gets larger.

The standard error of the mean measures the degree of accuracy of the sample mean (  ) as an estimate of the population mean (). It is also known as the standard deviation of the sampling distribution of the sampling mean, denoted by �.

Formula: � where: σ = population standard deviation n = sample size Remember that if we want to get a good estimate of the population mean, we have to make n sufficiently large. This fact is stated as a theorem in the Central Limit Theorem. Now, can you determine the standard error of the mean of the given set of data below? Your knowledge of the formula and manipulating the given data will be handy in solving this problem. I know that you can, so let’s do it! ACTIVITY Direction: Read the problems below then write your answer on a separate paper. 1. Determine the standard error of the mean for each of the following sample size n given the population standard deviation of 30. Round off your answer to the nearest hundredths. a. � = 5 b. � = 12 c. � = 28 d. � = 35 e. � = 40 2. Analyze the answers obtained in item number 1. What can you say about the relationship of the sample size and the standard error?

3. How does this relationship affect the distribution?

4. When do we obtain a good estimate of the mean?

5. When do we say that the mean is a poor estimate?

What is It  Central Limit theorem is important because it teaches researchers to use a limited sample to make intelligent and accurate conclusions about a greater population.  It also justifies the use of normal curve methods for a wide range of problems. � −



Furthermore, it justifies the use of the formula when computing for the probability that � J will take a value within a given range in the sampling distribution of J. where: � = is the sample mean µ = population mean, σ = population standard deviation n = sample size

When do you use these formulae?

J −



➢ �=  J −  ➢ � Study the illustrative sample problems below. 1. Assume that the variable is normally distributed, the average time it takes a group of senior high school students to complete a certain examination is 46.2 minutes while the standard deviation is 8 minutes. What is the probability that a randomly selected senior high school students will complete the examination in less than 43 minutes? Does it seem reasonable that a senior high school student would finish the examination in less than 43 minutes? a. If 50 randomly selected senior high school students take the examination, what is the probability that the mean time it takes the group to complete the test will be less than 43 minutes? Does it seem reasonable that the mean of the 50 senior high school students could be less than 43 minutes?

Solution for #1: Step1: Identify the parts of the problem. Given: = 46.2 � ������; � = 8 �������; J = 43 ������� Find: (J < 43) Step 2: Use the formula to find the z-score. J −  43 − 46.2 �= =  8 � = −�. ��

Step 3: Use the z-table to look up the z-score you calculated in step 2. � = −0.40 has a corresponding area of 0.1554.

0.1 5 54

Step 4: Draw a graph and plot the z-score and its corresponding area. Then, shade the part that you’re looking for: (�J < 43)

Since we are l will be on the left part of – 0.40.

ed part – 0.40

Step 5: Subtract your z-score from 0.500.

(J < 43) = 0.500 − 0.1554 �(J < 43) = 0.3446 Step 6: Convert the decimal in Step 5 to a percentage. (J < 43) = 34.46%  Therefore, the probability that a randomly selected senior high school student will complete the examination in less than 43 minutes is 34.46%. Yes, it is reasonable to finish the exam in less than 43 minutes since the probability is more than 1. Solution for #1.a: Step1: Identify the parts of the problem. = 46.2 �������; � = 8 �������; J = 43 �������; = 50 �������� Find: (J < 43)

Given:

Step 2: Use the formula to find the z-score. J −  43 − 46.2

� � = −�. �� Step 3: Use the z-table to look up the z-score you calculated in step 2. � = −2.83 has a corresponding area of 0.4977 Step 4: Draw a graph and plot the z-score and its corresponding area. Then, shade the part that you’re looking for: (�J < 43)

shaded part

0.4977

–2.83

Since we are looking for the probability less than 43 minutes, the shaded part will be on the left part of – 2.83. Step 5: Subtract your z-score from 0.500. (J < 43) = 0.500 − 0.4977 �(J < 43) = 0.0023 Step 6: Convert the decimal in Step 5 to a percentage. (J < 43) = 0.23%  Therefore, the probability that a randomly selected 50 senior high school students will complete the examination in less than 43 minutes is 0.23%. No, it’s not reasonable since the probability is less than 1.

2. An electrical company claims that the average life of the bulbs it manufactures is 1 200 hours with a standard deviation of 250 hours. If a random sample of 100 bulbs is chosen, what is the probability that the sample mean will be between 1150 hours and 1 250 hours? Solution: Step1: Identify the parts of the problem. Given:

= 1200 ℎ ; � = 250 ℎ; J = 1 150 & 1 250 ℎ����

� = 100 �����

Unknown: (1150 < J < 1250) Step 2: Use the formula to find the z-score. J −  1150 − 1200

1250 − 1200

�

�

� = −� �=� Step 3: Use the z-table to look up the z-score you calculated in step 2. � = ±2 has a corresponding area of 0.4772 Step 4: Draw a graph and plot the z-score and its corresponding area. Then, shade the part that you’re looking for: (1150 < � < 1250)

0.477 2

0.477 2

Since we are looking for the probability between 1 150 hours and 1 250 hours, the shaded part will be between –2 and 2. Step 5: Add the two z-score values. (1150 < J < 1250) = 0.4772 + 0.4772 �(1150 < J < 1250) = 0.9544 Step 6: Convert the decimal in Step 5 to a percentage. (1150 < J < 1250) = 95.44%  Therefore, the probability of randomly selected 100 bulbs to have a sample mean between 1 150 hours and 1 250 hours is 95.44%. What’s More

Let’s see how well you understood our discussion. At this point, I want you to solve the following problems. Show your complete solution by following the step-bystep procedure. 1. The average number of milligrams (mg) of cholesterol in a cup of a certain brand of ice cream is 660 mg, the standard deviation is 35 mg. Assume the variable is normally distributed. a. If a cup of ice cream is selected, what is the probability that the cholesterol content will be more than 670 mg? b. If a sample of 10 cups of ice cream is selected, what is the probability that the mean of the sample will be larger than 670 mg? 2. In a study of the life expectancy of 400 people in a certain geographic region, the mean age at death was 70 years, and the standard deviation was 5.1 years. If a sample of 50

people from this region is selected, what is the probability that the mean life expectancy will be less than 68 years? 3. The average cholesterol content of a certain canned goods is 215 milligrams, and the standard deviation is 15 milligrams. Assume that the variable is normally distributed. If a sample of 25 canned goods is selected, what is the probability that the mean of the sample will be greater than 220 milligrams? 4. The average public elementary school has 468 students with a standard deviation of 87. If a random sample of 38 public elementary schools is selected, what is the probability that the number of students enrolled is between 445 and 485? What I Have Learned

This time, I want you to work on the activity below based on your understanding of the topic by completing each sentence. I. Supply the missing words/phrase that will make the sentence complete. 1. Central Limit Theorem states that the sampling distribution of the mean approaches a ____________________________ as the sample size increases. 2. A good estimate of the mean is obtained if the standard error of the mean is small or _______________. 3. The mean is a poor estimate if the standard error of the mean is _______________. 4. The mean of the sampling distribution of the sample means is _______________ to the population mean.

5. The _____________________________ measures the degree of accuracy of the sample mean as an estimate of the population mean. II. In your own understanding, answer the questions below. 1. Do we always add or subtract from 0.50? Explain. 2. When do we add the corresponding area of the z-score to 0.50? 3. When do we add the two corresponding areas of the zscore?

II. Enumerate the steps in solving problem using Central Limit Theorem?

What I Can Do

Directions: Read, analyze, and solve the problems below. Show your complete solutions. 1. There are 250 dogs at a dog show that weigh an average of 12 pounds, with a standard deviation of 8 pounds. If 4 dogs are chosen at random, what is the probability that the average weight is greater than 8 pounds? 2. The average number of pages in a novel is 326 with a standard deviation of 24 pages. If a sample of 50 novels is randomly chosen, what is the probability that the average number of pages in these books is between 319 and 331? 3. The number of driving miles before a certain kind of tire begins to show wear is on the average, 16,800 miles with a standard deviation of 3,300 miles.

a. What is the probability that the 36 tires will have an average of less than 16,000 miles until the tires begin to wear out? b. What is the probability that the 36 tires will have an average of more than 18,000 miles until the tires begin to wear out? Assessment Directions: Read and analyze the story below. Write the letter of the correct answer on your answer sheet. 1. According to Central Limit theorem, which sample size will give a smaller standard error of the mean? A. 7

B. 12

C. 23

D. 40

2. If a population is not normally distributed, the distribution of the sample means for a given sample size n will ____________. A. be positively skewed. B. be negatively skewed. C. take the same shape as the population. D. approach a normal distribution as n increases. 3. 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 4. 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 5. What is the standard error of the mean if the sample size is 25 with standard deviation of 16? A. 6.25 B. 3.2 C. 1.25 D. 0.64

6. 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 7. In a study done on the 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 Distribution B. Central Limit Theorem Distribution

C.

Discrete D.

Probability Binomial

8. 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. 0.6% D. 99% 9. In a metal fabrication process, metal rods are produced that have an average length of 20.5 meters with a standard deviation of 2.3 meters. A quality control specialist collects a random sample of 30 rods and measures their lengths. Suppose the resulting sample mean is 19.5 meters. Which of the following statements is true? A. This sample mean is 2.38 standard deviations above what we expect. B. This sample mean is 2.38 standard deviations below what we expect. C. This sample mean is only 1 standard deviation above the population mean.

D. This sample mean is more than 3 standard deviations away from the population mean. For number 10-11, 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. 10. Compute the z-score for 6 hours of sleep. A. 1.04 B. 0.18 C. 0.52

D. 0.82

11. 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. 0.4286 For number 12-14, refer to the problem below. The amount of fuel used by jumbo jets to take off is normally distributed with a mean of 4, 000 gallons and a standard deviation of 125 gallons. A sample of 40 jumbo jets are randomly selected. 12. Compute the z-score for 3, 950 gallons. A. – 0.4 B. 0.4 C. – 2.53

D. 2.53

13. What is the probability that the mean number of gallons of fuel needed to take off for a randomly selected sample of 40 jumbo jets will be less than 3, 950 gallons? a. 78.1% B. 34.5% C. 2.5% D. 0.57% 14. What is the probability that the mean number of gallons of fuel needed to take off for a randomly selected sample of 40 jumbo jets will be more than 3, 950 gallons? b. 0.57% B. 49.43% C. 65.54% D. 99.43%

15. Researchers found that boys playing high school football recorded an average of 355 hits to the head with a standard deviation of 80 hits during a season. What is the probability on a randomly selected team of 48 players that the average number of head hits per player is between 340 and 360? A. 56.96% B. 43.04% C. 40.32% D. 16.64% Additional Activities

Direction. Use a separate sheet of paper to answer the activity below. J urnal Writing: In your own words, explain the usefulness o of Central Limit Theorem (CLT) in solving problems involving sampling. Also, cite other importance of ...