Question

Imagine that there are 100 different researchers each studying the sleeping habits of college freshmen. Each researcher takes a random sample of size 50 from the same population of freshmen. Each researcher is trying to estimate the mean hours of sleep that freshmen get at night, and each one constructs a \(95 \%\) confidence interval for the mean. Approximately how many of these 100 confidence intervals will NOT capture the true mean? a. None b. 1 or 2 c. 3 to 7 d. about half e. 95 to 100 f. other

Short answer

c. 3 to 7

Step-by-step solution

Understanding Confidence Intervals

A confidence interval gives a range of values for the true population parameter, with a specified level of confidence, here 95%. This means if we were to take many samples and construct their confidence intervals, approximately 95% of those intervals would contain the true mean.

Calculating the Expected Number of Intervals Not Capturing the True Mean

Out of every 100 confidence intervals constructed, 95% would capture the true mean. Therefore, 5% might not capture it. In this case, 5% of 100 is equal to 5.

Interpreting the Results

Thus, approximately 5 out of 100 confidence intervals will not capture the true mean, as 95% of them will capture it. This corresponds to the range of possible answers given in the exercise choices.

Key concepts

Sampling Distribution

When researchers take a sample from a population and calculate a statistic, like the sample mean, this statistic can vary from sample to sample. This variation is known as a sampling distribution. Imagine taking repeated random samples from a population of college freshmen and calculating the mean hours of sleep for each sample. Even though each sample is randomly chosen from the same population, the sample means will differ slightly. The collection of these sample means creates what we call the sampling distribution of the sample mean.

The sampling distribution is crucial because it allows us to understand how the sample mean behaves. It generally follows a normal distribution, especially if the sample size is large, due to the Central Limit Theorem. This makes it possible to use statistical methods to make inferences about the population mean.

• Sampling distributions provide the foundation for constructing confidence intervals.
• They help in estimating the variability and reliability of the sample statistics.

Understanding sampling distributions allows researchers to make more accurate inferences about the population.

Population Mean

The population mean is a fundamental concept in statistics, representing the average of a certain characteristic for every individual in a population. In the context of the sleeping habits of freshmen, it’s the mean number of hours all freshmen sleep at night. Researchers rarely have access to data for an entire population due to time and resource constraints. Thus, they rely on samples to estimate the population mean.

When researchers take samples and calculate the mean of these samples, they use this information to infer the population mean. It is important to remember that the sample mean serves as an estimator of the population mean, but it is not always exactly equal to the population mean. This is due to the natural variability present in any sample.

• Estimating the population mean is a primary goal in many research studies.
• It's used to make broader conclusions and decisions about the population.

Therefore, the sample mean is used in conjunction with the sampling distribution to estimate and infer the population mean with a known level of confidence.

Statistical Inference

Statistical inference is the process of drawing conclusions about a population based on information from a sample. It involves estimating population parameters and testing hypotheses. In the exercise, each researcher’s goal is to infer the mean number of sleep hours freshmen get per night based on sample data.

One primary method of inference is constructing confidence intervals, which provide a range of likely values for the population parameter. To apply statistical inference, researchers must assume that the sample is representative of the population.

• Statistical inference includes both estimation (like confidence intervals) and hypothesis testing.
• It allows researchers to use sample data to make more general conclusions about the population.

By understanding and applying statistical inference, researchers can confidently estimate and draw conclusions about population parameters, even when the true values are unknown.

Confidence Level

The confidence level is associated with the probability that a confidence interval, calculated from a sample, will contain the true population parameter. In the problem scenario, researchers use a 95% confidence level to estimate the mean number of hours freshmen sleep at night. This means that if 100 different samples were taken and a confidence interval was calculated for each, 95 of those intervals would be expected to contain the true population mean.

A 95% confidence level, therefore, suggests a relatively high level of certainty in the estimation process. However, it's important to understand that a higher confidence level results in wider intervals around the sample mean.

• A higher confidence level indicates greater certainty that the interval includes the true mean.
• The choice of confidence level affects the width of the confidence interval.

By understanding confidence levels, you’ll appreciate the balance between certainty and precision in statistical estimations.