Defining sampling distribution of the sample mean for.pptx

G11 Math

1. Defining sampling distribution of the sample mean for normal population when the variance is: a) known b) unknown

2. Activity 1: KNOWING MO NA? Instruction: Read and analyze the following situations. Determine whether the following statements have a known or unknown population variance. Write your answer on the space provided. Identify also the formula to be used to estimate the standard error of the mean by writing the symbol when the population variance is known and the symbol when the population variance is unknown  

3. Situation Answer Standard Error Formula 1. Because of Inclusive Education, learners with disabilities are also part of the normal students. Consider a population of the PWD learners consisting of 1, 2, 3, 4, and 5. Samples of size 2 are drawn from this population 2. Given 𝜇 = 15, 𝑠 = 6 𝑎𝑛𝑑 𝑛 = 16

4. Situation Answer Standard Error Formula 3. IQ scores are normally distributed with standard deviation 𝜎 = 15 𝑝𝑜𝑖𝑛𝑡𝑠. You wish to determine the 𝜇 IQ of all students in Matiyaga National High School by taking a random sample of 16 students.

5. Situation Answer Standard Error Formula 4. The general weighted average (GWA) of the Grade 7 applicants for admission in QSHS over the past 7 years has been consistently 𝜇 = 92.78. This year a sample data of the applicant’s GWA are 98, 89, 92, 95, 96, 90, 91, 92, 97 and 93

6. Situation Answer Standard Error Formula 5. A finite population composed of six items whose values are 2, 3, 6, 7, 8, and 10. Samples of 3 items are drawn at random without replacement.

7. Situation Answer Standard Error Formula 1. Because of Inclusive Education, learners with disabilities are also part of the normal students. Consider a population of the PWD learners consisting of 1, 2, 3, 4, and 5. Samples of size 2 are drawn from this population Known – This is a population data. Although the value of the variance is not given, you can still determine the population mean and population variance of the data using the formulas , 2. Given 𝜇 = 15, 𝑠 = 6 𝑎𝑛𝑑 𝑛 = 16 Unknown - The population variance is unknown. The given values are the population mean 𝜇 = 15, the sample standard deviation 𝑠 = 6, and the sample size 𝑛 = 16. Situation Answer Standard Error Formula 1. Because of Inclusive Education, learners with disabilities are also part of the normal students. Consider a population of the PWD learners consisting of 1, 2, 3, 4, and 5. Samples of size 2 are drawn from this population 2. Given 𝜇 = 15, 𝑠 = 6 𝑎𝑛𝑑 𝑛 = 16 Unknown - The population variance is unknown. The given values are the population mean 𝜇 = 15, the sample standard deviation 𝑠 = 6, and the sample size 𝑛 = 16.

8. Situation Answer Standard Error Formula 3. IQ scores are normally distributed with standard deviation 𝜎 = 15 𝑝𝑜𝑖𝑛𝑡𝑠. You wish to determine the 𝜇 IQ of all students in Matiyaga National High School by taking a random sample of 16 students. Known - the population variance is given with the value of the square of the standard deviation Situation Answer Standard Error Formula 3. IQ scores are normally distributed with standard deviation 𝜎 = 15 𝑝𝑜𝑖𝑛𝑡𝑠. You wish to determine the 𝜇 IQ of all students in Matiyaga National High School by taking a random sample of 16 students.

9. Situation Answer Standard Error Formula 4. The general weighted average (GWA) of the Grade 7 applicants for admission in QSHS over the past 7 years has been consistently 𝜇 = 92.78. This year a sample data of the applicant’s GWA are 98, 89, 92, 95, 96, 90, 91, 92, 97 and 93 The population variance is unknown. The only given values are the population mean 𝜇 = 92.78 and the sample size 𝑛 = 10. Situation Answer Standard Error Formula 4. The general weighted average (GWA) of the Grade 7 applicants for admission in QSHS over the past 7 years has been consistently 𝜇 = 92.78. This year a sample data of the applicant’s GWA are 98, 89, 92, 95, 96, 90, 91, 92, 97 and 93 The population variance is unknown. The only given values are the population mean 𝜇 = 92.78 and the sample size 𝑛 = 10.

10. Situation Answer Standard Error Formula 5. A finite population composed of six items whose values are 2, 3, 6, 7, 8, and 10. Samples of 3 items are drawn at random without replacement. Known – This is a population data; you can determine the population mean and variance of the data using the formulas , Situation Answer Standard Error Formula 5. A finite population composed of six items whose values are 2, 3, 6, 7, 8, and 10. Samples of 3 items are drawn at random without replacement.

11. Distribution of the Sample Mean for Normal Population 1. Population variance is known The population has a mean μ and variance of , the distribution of the sample mean is (at least approximately) normal and standard error of the mean , where σ is the population standard deviation and 𝑛 is the sample size. To determine the probability of a certain event, we can use the 𝑧 − 𝑑𝑖𝑠𝑡𝑟𝑖𝑏𝑢𝑡𝑖𝑜𝑛 by transforming the mean of the sample data to an approximately normal variable 𝑧 , using the relation . This distribution is best applied for large sample sizes, say 𝑛 ≥ 30.  

12. 2 . Population variance is unknown The standard error of the mean becomes , where 𝑠 is the point estimate of 𝜎 (population standard deviation) or the sample standard deviation and 𝑛 is the sample size. To estimate the population parameters, we can use the 𝑡 − 𝑑𝑖𝑠𝑡𝑟𝑖𝑏𝑢𝑡𝑖𝑜𝑛 by using the formula . Remember that as n the sample size is very large, the standard deviation 𝑠 is almost indistinguishable from the population standard deviation 𝜎 and therefore 𝑡 and 𝑧 distributionsare essentially identical. Remember that, we use the 𝑡 − 𝑑𝑖𝑠𝑡𝑟𝑖𝑏𝑢𝑡𝑖𝑜𝑛 for smallsample size, say 𝑛 < 30.  

13. Independent Activity Instruction: Read and analyze the following situations and supply the values of the following variables (if there is any). On the third column, write known on the space provided if the situation gives or can compute the value of the variance, otherwise write unknown . Identify also the formula to be used to estimate the standard error of the mean by writing the symbols 𝜎𝑥̅ , when the population variance is known and 𝑠𝑥̅ when the population variance is unknown.