Refer to Exercise 6.5. Assume that a random sample of n=2...

Refer to Exercise 6.5. Assume that a random sample of $n=2$ measurements is randomly selected from the population. a. List the different values that the sample median $M$ may assume, and find the probability of each. Then give the sampling distribution of the sample median. b. Construct a probability histogram for the sampling distribution of the sample median, and compare it with the probability histogram for the sample mean (Exercise 6.5 , part b).

Refer to Exercise 6.5. Assume that a random sample of $n=2$ measurements is randomly selected from the population.
a. List the different values that the sample median $M$ may assume, and find the probability of each. Then give the sampling distribution of the sample median.
b. Construct a probability histogram for the sampling distribution of the sample median, and compare it with the probability histogram for the sample mean (Exercise 6.5 , part b).

Key Concepts

Probability Histogram

A probability histogram is a graphical representation that displays the probability or relative frequency of each outcome in a discrete probability distribution. It is used to visualize the spread and shape of the sampling distribution for a statistic, making it easier to compare distributions such as those of the sample mean and sample median.

Sample Mean

The sample mean is calculated as the sum of the observations divided by the number of observations. Its sampling distribution is a central concept in statistics, especially because of the central limit theorem, which states that the distribution of the sample mean approaches a normal distribution as the sample size increases.

Random Sampling

Random sampling is the process of selecting a subset of individuals from a population in such a way that every possible sample of the specified size has an equal chance of being selected. It ensures that the sampling distribution accurately reflects the properties of the overall population, which is crucial for valid statistical inference.

Sampling Distribution

This concept refers to the distribution of a statistic (such as the mean or median) computed from a large number of samples drawn from a population. It provides insights into the variability and behavior of the statistic and is fundamental for making inferences about the population parameters.

Sample Median

The sample median is the middle value in a data set when it is ordered. In the context of sampling distributions, understanding the distribution of the sample median helps in assessing its reliability and variability as an estimator of the population median, particularly in small samples.

Transcript

00:01 So for this question, the population is 4, 5 and 9.

00:19 4 .5 .9.

00:25 And the sample size is 2 because you're picking to a random.

00:31 So n equals to 2.

00:47 And then the first question wants you to figure out, the first question wants to figure out the value of the population median.

00:58 And mind you the median is always the median number when you arrange the data ascending order and from four five and nine you can definitely tell that nine is the middle number so definitely the population median the population median is equal to five and that's the answer the second question you do really and remember that the two values are randomly selected of replacement from 457 and we're going to write the samples of the possible um the possible son the samples of the possible i'm going to write the samples of the possible outcomes in order to describe the sampling distribution of the sampling name so the samples the samples would be 4 .4 4 .5 4 .9 5 .4 5 .5 and 5 .9.

02:58 Also, 9 .4, 9 .5 and 9 .9.

03:10 So there are 9 probabilities, total probabilities equals to 9.

03:20 Right.

03:22 And then to construct the table for sampling distribution of median, we need to find.

03:27 The sample media in order to construct the table and you know that number of probability is equal to nine and then in order to find the sample media we're going to write the sample over here and then we're going to have the samples the nine samples which is 4 .4 4 .5 4 .9 5 .4 5 .4 5 .5 5 .5 5 .5 5 also 9 .4, 9 .5 and 9 .9.

04:27 And then we're going to find the sample median for each of them in order to finish the table.

04:35 So the sample median.

04:52 And we know that the median is the sum of the number is divided by the actual number, the total numbers we have.

05:03 And since we have two figures, because we have two um we're picking two numbers at random so it's two and then we have like two samples for every sample 4 plus 4 divided by 2 is going to be 4 so that's going to be the sample sample median 4 plus 5 they're divided by 2 is going to be 4 .5 that's going to be the sample median the 4 .9 4 .4 plus 9 divided by 2 is going to be 6 points that's a sample median high cost 4 by 2 is 4 .5.

05:43 This one is 5.

05:46 This one is 7.

05:50 This one is 6 .5.

05:54 This one is 7.

05:57 And this one is 9...

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