In Exercises 7–10, use the same population of {4, 5, 9} that was ...

8. Sampling Distributions & Confidence Intervals: Proportion

Sampling Distribution of Sample Proportion

Statistics

8. Sampling Distributions & Confidence Intervals: Proportion

Sampling Distribution of Sample Proportion

1:18 minutes

Problem 6.3.9a

Textbook Question

In Exercises 7–10, use the same population of {4, 5, 9} that was used in Examples 2 and 5. As in Examples 2 and 5, assume that samples of size n = 2 are randomly selected with replacement.

Sampling Distribution of the Sample Median

a. Find the value of the population median.

Verified step by step guidance

Step 1: Understand the concept of the median. The median is the middle value of a data set when the values are arranged in ascending order. If the number of values is odd, the median is the middle value. If the number of values is even, the median is the average of the two middle values.

Step 2: Arrange the population values {4, 5, 9} in ascending order. Since the population is already sorted, no further action is needed.

Step 3: Determine the number of values in the population. Here, the population consists of three values: 4, 5, and 9.

Step 4: Identify the middle value of the population. Since the population size is odd (n = 3), the median is the middle value of the sorted list, which is the second value.

Step 5: Conclude that the population median is the middle value of the sorted population, which can be directly identified from the ordered list.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Population Median

The population median is the middle value of a dataset when it is ordered from least to greatest. For an odd number of observations, it is the central number, while for an even number, it is the average of the two central numbers. In the context of the given population {4, 5, 9}, the median is calculated by ordering the values and identifying the middle point.

Sampling Distribution

A sampling distribution is the probability distribution of a statistic (like the median) obtained from a large number of samples drawn from a specific population. It illustrates how the statistic varies from sample to sample. Understanding the sampling distribution is crucial for making inferences about the population based on sample data, especially when samples are taken with replacement.

Random Sampling with Replacement

Random sampling with replacement means that each time an item is selected from the population, it is returned before the next selection. This method ensures that each selection is independent and that the same item can be chosen multiple times. It is important for maintaining the integrity of the sampling process and for accurately estimating the sampling distribution of statistics like the median.

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In Exercises 7–10, use the same population of {4, 5, 9} that was used in Examples 2 and 5. As in Examples 2 and 5, assume that samples of size n = 2 are randomly selected with replacement.Sampling Distribution of the Sample Mediana. Find the value of the population median.