Question

Suppose you know that the distribution of sample proportions of fifth grade students in a large school district who read below grade level in samples of 100 students is normal with a mean of 0.30 and a standard deviation of . you select a sample of 100 fifth grade students from this district and find that the proportion who read below grade level in the sample is . this sample proportion lies standard deviations above the mean of the sampling distribution. what is the probability that a second sample would be selected with a proportion greater than ​? round to three decimal places as needed.

Answer 1

We identify a problem as a proportion problem when dealing with categorical outcomes labeled as 'Success' or 'Failure.' Sample proportions can be normally distributed if sample size is large enough, and hypothesis testing involves comparing p-values to predetermined significance levels.

Step-by-step explanation:

When we talk about the distribution of sample proportions, it involves identifying the frequency of two possible outcomes, commonly referred to as 'Success' or 'Failure.' This is clearly a proportion problem because it pertains to categorical data. For instance, questions like 'What proportion of the population will vote for candidate A?' are examples of proportion problems.

The standard deviation of the sample proportions (P') is calculated using the formula \(\sigma_{p'} = \sqrt{\frac{p \cdot q}{n}}\), where p is the population proportion, q is 1 - p, and n is the sample size. According to the central limit theorem for proportions, if the sample size is sufficiently large, the sampling distribution of P' will be approximately normal.

The empirical rule states that about 95% of the time, a sample mean will fall within two standard deviations of the population mean. In hypothesis testing, if the p-value is lower than the significance level (\(\alpha\)), we reject the null hypothesis. This level could be 0.05, 0.01, or any other threshold determined before the test.