A random sample of 1000 students at a large college included...

A random sample of 1000 students at a large college included 428 who had one or more credit cards. For this sample, $\hat{p}=\frac{428}{1000}=0.428 .$ If another random sample of 1000 students from this university were selected, would you expect that $\hat{p}$ for that sample would also be 0.428 ? Explain why or why not.

Key Concepts

Sampling Distribution of a Proportion

The sampling distribution of a proportion represents the probability distribution of all possible values of the sample proportion that could be observed from repeated samples of the same size, drawn from the same population. This concept explains that while the expected value (mean) of the sampling distribution will be the true population proportion, individual sample proportions will vary around this mean due to random sampling variability.

Random Sampling

Random sampling is a method of selecting a subset of individuals in such a way that every individual in the population has an equal chance of being included. This technique is key in ensuring that the sample is representative of the overall population, thereby making inferences drawn from the sample more reliable.

Sample Proportion

The sample proportion is a statistic that estimates the probability of an outcome (in this case, having at least one credit card) based on the number of successes observed in a random sample. It is calculated by dividing the number of individuals with the attribute by the total number of individuals sampled.

Sampling Variability

Sampling variability refers to the natural fluctuations in sample statistics, such as the sample proportion, that occur from one random sample to another. Because each sample is different, the calculated proportion can vary, even when the underlying population parameter remains constant.

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