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 Proportion

c. Find the mean of the sampling distribution of the sample proportion of odd numbers.

검증된 단계별 안내

Step 1: Identify the population and the parameter of interest. The population is {4, 5, 9}, and we are interested in the proportion of odd numbers in the samples. Odd numbers in the population are {5, 9}.

Step 2: Determine the sample size (n = 2) and note that sampling is done with replacement. This means each sample can include repeated values from the population.

Step 3: List all possible samples of size 2 that can be drawn with replacement from the population. For example, the samples are {(4, 4), (4, 5), (4, 9), (5, 4), (5, 5), (5, 9), (9, 4), (9, 5), (9, 9)}.

Step 4: For each sample, calculate the sample proportion of odd numbers. For example, in the sample (4, 5), the proportion of odd numbers is 1/2 because only one of the two numbers is odd.

Step 5: Compute the mean of the sampling distribution of the sample proportion by averaging all the sample proportions calculated in Step 4. Use the formula:

μ = \frac{Σ p}{N}

, where Σp is the sum of all sample proportions and N is the total number of samples.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Sampling Distribution

A sampling distribution is the probability distribution of a statistic obtained from a larger population, created by taking multiple samples. In this context, it refers to the distribution of the sample proportion of odd numbers when samples of size n = 2 are drawn from the population {4, 5, 9}. Understanding this concept is crucial for analyzing how sample statistics behave and vary from the population parameter.

Sample Proportion

The sample proportion is the ratio of the number of successes (in this case, odd numbers) to the total number of observations in the sample. For the population {4, 5, 9}, the odd numbers are 5 and 9. The sample proportion helps in estimating the likelihood of finding odd numbers in random samples, which is essential for calculating the mean of the sampling distribution.

Mean of the Sampling Distribution

The mean of the sampling distribution of a sample proportion is the expected value of that proportion across all possible samples. It can be calculated as the population proportion of odd numbers. This mean provides insight into the central tendency of the sample proportions, allowing for predictions about the behavior of samples drawn from the population.