The candy machine Suppose a large candy machine has 45%...

The candy machine Suppose a large candy machine has 45$\%$ orange candies. Imagine taking an SRS of 25 candies from the machine and observing the sample proportion $\hat{p}$ of orange candies. (a) What is the mean of the sampling distribution of $\hat{p} ?$ Why? (b) Find the standard deviation of the sampling distribution of $\hat{p}$ . Check to see if the 10$\%$ condition is met. (c) Is the sampling distribution of $\hat{p}$ approximately Normal? Check to see if the Normal condition is met. (d) If the sample size were 50 rather than $25,$ how would this change the sampling distribution of $\hat{p} ?$

The candy machine Suppose a large candy machine has 45$\%$ orange candies. Imagine taking an SRS of 25 candies from the machine and observing the sample proportion $\hat{p}$ of orange candies.
(a) What is the mean of the sampling distribution of $\hat{p} ?$ Why?
(b) Find the standard deviation of the sampling distribution of $\hat{p}$ . Check to see if the 10$\%$ condition is met.
(c) Is the sampling distribution of $\hat{p}$ approximately Normal? Check to see if the Normal condition is met.
(d) If the sample size were 50 rather than $25,$ how would this change the sampling distribution of $\hat{p} ?$

Key Concepts

Effect of Sample Size on the Sampling Distribution

Increasing the sample size decreases the standard error, making the sampling distribution more narrowly concentrated around the population proportion. This results in more precise estimates of the population parameter, although the mean of the sampling distribution remains unchanged.

Normal Condition for Proportions

For the sampling distribution of the sample proportion to be approximately Normal, both np and n(1-p) must be sufficiently large, usually at least 10. This condition is important as it justifies the use of Normal approximations in hypothesis testing and confidence interval estimation.

10% Condition

This condition requires that the sample size be no more than 10% of the population if sampling without replacement. It ensures independence of observations, which is a necessary assumption for many statistical methods, including the use of the standard error formula without adjustment.

Sampling Distribution of a Sample Proportion

This is the probability distribution of the sample proportion (p-hat) obtained from repeatedly taking samples from a population. It describes how the observed proportions vary from sample to sample and is fundamental in statistical inference, especially when estimating population parameters.

Mean of the Sampling Distribution

The mean of the sampling distribution of a sample proportion is equal to the population proportion (p). This concept shows that, on average, the sample proportion is an unbiased estimator of the true proportion in the population.

Standard Error of the Sampling Distribution

The standard error quantifies the variability of the sample proportion and is calculated as the square root of the product of the population proportion and its complement divided by the sample size (?(p(1-p)/n)). It is crucial for understanding the precision of the sample estimate and for constructing confidence intervals.

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