Use the same population of {4,5,9} that was used in Examples...

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 Proportion a. For the population, find the proportion of odd numbers. b. Table $6-2$ describes the sampling distribution of the sample mean. Construct a similar table representing the sampling distribution of the sample proportion of odd numbers. Then combine values of the sample proportion that are the same, as in Table $6-3 .$ (Hint: See Example 2 on page 258 for Tables $6-2$ and $6-3,$ which describe the sampling distribution of the sample mean.) c. Find the mean of the sampling distribution of the sample proportion of odd numbers. d. Based on the preceding results, is the sample proportion an unbiased estimator of the population proportion? Why or why not?

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 Proportion
a. For the population, find the proportion of odd numbers.
b. Table $6-2$ describes the sampling distribution of the sample mean. Construct a similar table representing the sampling distribution of the sample proportion of odd numbers. Then combine values of the sample proportion that are the same, as in Table $6-3 .$ (Hint: See Example 2 on page 258 for Tables $6-2$ and $6-3,$ which describe the sampling distribution of the sample mean.)
c. Find the mean of the sampling distribution of the sample proportion of odd numbers.
d. Based on the preceding results, is the sample proportion an unbiased estimator of the population proportion? Why or why not?

Key Concepts

Sample Proportion

The sample proportion is a statistic that estimates the population proportion by calculating the fraction of sample members that have a certain attribute. It is obtained by dividing the number of favorable outcomes in the sample by the sample size. This statistic is used as an estimator for the population proportion and its properties, such as its sampling distribution, are crucial in assessing the accuracy of the estimation.

Sampling Distribution

A sampling distribution is the probability distribution of a statistic—such as a sample mean or sample proportion—calculated from all possible samples of a fixed size drawn from a population. It provides a framework to understand the variability of the statistic from sample to sample and is fundamental in making inferences about the population parameter.

Unbiased Estimator

An estimator is considered unbiased if its expected value equals the true parameter it is intended to estimate. In the context of sampling distributions, an unbiased estimator will, on average, reflect the actual population value when repeated samples are taken. This property is critical in ensuring that statistical inferences drawn from the sample are valid representations of the population parameter.

Population Proportion

This concept refers to the fraction or percentage of members in a population that possess a particular attribute. It is computed by dividing the number of individuals with the characteristic by the total number of individuals in the population. The population proportion is a fixed value for the entire set and serves as the parameter of interest in many inferential statistics procedures.

Sampling with Replacement

Sampling with replacement involves selecting members from a population where each member is returned to the population before the next selection. This process ensures that each draw is independent of the previous ones and that the probability of selecting any specific member remains constant across samples. It plays a crucial role in forming the sampling distribution when the selection process allows for repeated inclusion of the same member.

Transcript

00:02 So this question, the population is 4, 5, 9.

00:23 And then the sample size is 2.

00:28 Because 2 are pitted, 2 room samples are pitted without replacements.

00:34 The size is 2.

00:37 And then the proportion of, the question says that two are picked randomly.

00:44 And the first question wants us to find the proportion.

00:49 Of odd numbers.

00:54 So the proportion of all numbers is going to be from the proportion of odd numbers from four five and nine is going to be two which is five and nine because four is an evil number and the next question the next question wants us to describe to construct a similar table a table that describes the sampling distribution of the sample mean and represents the sampling distribution of the sample proportion of all numbers.

01:50 So to start we're going to have to put the samples right here.

02:03 So the table we're going to start with the samples.

02:14 And we know that the samples, if two are picked around only 4 .4, 4 .5, 4 .9, 5, 5, 5, 5, 5, 5 .5, 5.

02:37 5.

02:39 5 .9, 9 .4, 9 .5, 9 .9 .9.

02:56 And then the other side is going to be the proportion of old numbers...

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