Sampling Distribution Estimation Hypothesis Testing Chapter 2 Sampling Distribution Estimation Hypothesis Testing

Sampling Distributions Introduction Sampling Distribution of Sample Mean ( ) Sampling Distribution of Sample Proportion ( )

Introduction A sampling distribution - probability distribution of a sample statistic based on all possible simple random sample of the same size from the same population. If we take several sample and find mean of the sample, therefore the distribution of the sample mean called sampling distribution of the sample mean, . For example, suppose you sample 50 students from your college regarding their mean GPA. If you obtained many different samples of 50, you will compute a different mean for each sample. We are interested in the distribution of all potential mean GPA we might calculate for any given sample of 50 students.

Introduction If we take several sample and find the ratio of the specific characteristic in the sample, therefore the distribution of the sample proportion called sampling distribution of the sample proportion, .

Introduction The reason we select a sample is to collect data to answer a research question about a population. The sample results provide only estimates of the values of the population characteristics. The reason is simply that the sample contains only a portion of the population. With proper sampling methods, the sample results can provide “good” estimates of the population characteristics.

Sampling Distribution of the Sample Mean ( ) The probability distribution of sample is called sampling distribution. It lists the various values that can assume, and the probability of each value of . If the population is normally distributed with mean (μ) and standard deviation (σ), the sampling distribution of the sample mean, is also normally distributed with: i) Mean iii) Standard deviation / standard error ii) Variance where n is the sample size

Sampling Distribution of the Sample Mean ( ) Z-value for the sampling distribution of mean ( ):

The Central Limit Theorem If all samples of a particular size are selected from any population, the sampling distribution of the sample mean is approximately a normal distribution. This approximation improves with larger samples.

Sample Mean Sampling Distribution: If the Population is not Normal If the population is not normally distributed, apply central limit theorem. Central Limit Theorem: Even if the population is not normal, sample means from the population will be approximately normal as long as the sample size is large enough (n≥30). n↑ the sampling distribution becomes almost normal regardless of shape of population As the sample size gets large enough…

Properties and Shape of Sampling Distribution of the Sample Mean ( ) If n ≥ 30 (large) , the sampling distribution of the sample mean is normally distributed; Note: If the unknown then it is estimated by . If n < 30 (small), is known, and the sampling distribution of the sample mean is normally distributed if the sample is from the normal population;

If n<30 and is unknown. t distribution with n-1 degree of freedom is use;

Example: Suppose a population has mean μ = 8 and standard deviation σ = 3. A random sample of size n = 36 is selected. What is the probability that the sample mean is between 7.8 and 8.2? Solution: Even if the population is not normally distributed, the central limit theorem can be used (n > 30). So the sampling distribution of the sample mean is approximately normal with mean, and standard deviation, Hence, Z

Example: The amount of time required to change the oil and filter of any vehicles is normally distributed with a mean of 45 minutes and a standard deviation of 10 minutes. A random sample of 16 cars is selected. What is the standard error of the sample mean to be? What is the probability of the sample mean between 45 and 52 minutes? What is the probability of the sample mean between 39 and 48 minutes? Find the two values between the middle 95% of all sample means.

Solution:

Z

Exercise: A certain type of thread is manufactured with a mean tensile strength is 78.3kg, and a standard deviation is 5.6kg. Assuming that the strength of this type of thread is distributed approximately normal, find: a) The probability that the mean strength of a random sample of 10 such thread falls between 77kg and 78kg. b) The probability that the mean strength greater than 79kg. c) The probability that the mean strength is less than 76kg. d) The value of X to the right of which 15% of the mean computed from random samples of size 10 would fall.

Sampling Distribution of the Sample Proportion ( ) The probability distribution of the sample proportion ( ) is called its sampling distribution. The population and sample proportion are denoted by p and respectively, and calculated as: where N = total number of elements in the population; X = number of elements in the population that possess a specific characteristic; n = total number of elements in the sample; and x = number of elements in the sample that possess a specific characteristic and

For the large values of n (n ≥ 30), the sampling distribution is very closely normally distributed. Mean and Standard Deviation of Sample Proportion

Example: If the true proportion of voters who support Proposition A is p = 0.40, what is the probability that a sample of size 200 yields a sample proportion between 0.40 and 0.45? Solution: Z

Example The National Survey of Engagement shows about 87% of freshmen and seniors rate their college experience as “good” or “excellent”. Assume this result is true for the current population of freshmen and seniors. Let be the proportion of freshmen and seniors in a random sample of 900 who hold this view. Find the mean and standard deviation. Let p the proportion of all freshmen and seniors who rate their college experience as “good” or “excellent”. Then, p = 0.87 and q = 1 – p = 1 – 0.87 = 0.13 The mean of the sample distribution of is: The standard deviation of is: Solution: