Notation Population Sample Sampling Math 140 Distribution Introductory Statistics µ µ Mean x x Standard Professor Bernardo Ábrego Deviation σ s σ x Lecture 16 Sections 7.1,7.2 Size N n Properties of The Sampling Example 1 Distribution of The Sample Mean The mean µ x of the sampling distribution of x equals the mean of the population µ: Problems usually involve a combination of the µx = µ three properties of the Sampling Distribution of the Sample Mean, together with what we The standard deviation σ x of the sampling distribution of x , also called the standard error of the mean, equals the standard learned about the normal distribution. deviation of the population σ divided by the square root of the sample size n: σ Example: Average Number of Children σ x = n What is the probability that a random sample The Shape of the sampling distribution will be approximately of 20 families in the United States will have normal if the population is approximately normal; for other populations, the sampling distribution becomes more normal as an average of 1.5 children or fewer? n increases. This property is called the Central Limit Theorem. 1 Example 1 Example 1 Example: Average Number Number of Children Proportion of families, P(x) of Children (per family), x µx = µ = 0.873 What is the probability that a 0 0.524 random sample of 20 1 0.201 σ 1.095 σx = = = 0.2448 families in the United States 2 0.179 n 20 will have an average of 1.5 3 0.070 children or fewer? 4 or more 0.026 Mean (of population) 0.6 µ = 0.873 0.5 0.4 Standard Deviation 0.3 σ =1.095 0.2 0.1 0.873 0 01234 Example 1 Example 2 µ = µ = 0.873 Find z-score of the value 1.5 Example: Reasonably Likely Averages x x − mean z = = What average numbers of children are σ 1.095 SD σ = = = 0.2448 reasonably likely in a random sample of 20 x x − µ 1.5 − 0.873 n 20 = x = families? σx 0.2448 ≈ 2.56 Recall that the values that are in the middle normalcdf(−99999,2.56) ≈ .9947 95% of a random distribution are called So in a random sample of 20 Reasonably Likely. families there is a 99.47% 0.873 probability that the mean number of children per family will be less than 1.5 2 Finding Probabilities for Example 2 Sample Totals Example: Reasonably Likely Averages Sometimes situations are stated in terms of the total number in the sample rather than the average number: “What is the What average numbers of children are probability that there are 30 or fewer children in a random sample of 20 families in the United States?” You have the reasonably likely in a random sample of 20 choice of two equivalent ways to do this problem. families? Method I: Find the equivalent average number of children, x, by dividing the total number of children, 30, by the sample size, 20: 30 x = =1.5 Recall that the values that are in the middle 20 95% of a random distribution are called Then you can use the same formulas and procedure as in the Reasonably Likely. previous examples. Method II: Convert the formulas from the previous examples to Note that by calculating the z-scores of 2.5% and 97.5% we find that equivalent formulas for the sum, then proceed as in the next the Reasonably Likely values are those values within 1.96 standard example. deviations from the mean. That is, between µ – 1.96 σ and µ + 1.96 σ Sampling Distribution Examples 3 and 4 of the Sum of a Sample If a random sample of size n is selected with mean µ Ex3: The Probability of 25 or fewer Children and standard deviation σ, then the mean of the sampling distribution of the sum is What is the probability that a random sample of 20 families in the United States will have a µsum = nµ total of 25 children or fewer? the standard error of the sampling distribution of the sum is σ sum = n ⋅σ Ex4: Reasonably Likely Totals the shape of the sampling distribution will be In a random sample of 20 families, what total approximately normal if the population is approximately normal; for other populations, the sampling distribution numbers of children are reasonably likely? becomes more normal as n increases. Note: To get the “sum” formulas just multiply by n 3.