Question

Student Misinterpretations Suppose that a student is working on a statistics project using data on pulse rates collected from a random sample of 100 students from her college. She finds a \(95 \%\) confidence interval for mean pulse rate to be (65.5,71.8) . Discuss how each of the statements below would indicate an improper interpretation of this interval. (a) I am \(95 \%\) sure that all students will have pulse rates between 65.5 and 71.8 beats per minute. (b) I am \(95 \%\) sure that the mean pulse rate for this sample of students will fall between 65.5 and 71.8 beats per minute. (c) I am 95\% sure that the confidence interval for the average pulse rate of all students at this college goes from 65.5 to 71.8 beats per minute. (d) I am sure that \(95 \%\) of all students at this college will have pulse rates between 65.5 and 71.8 beats per minute. (e) I am \(95 \%\) sure that the mean pulse rate for all US college students is between 65.5 and 71.8 beats per minute. (f) Of the mean pulse rates for students at this college, \(95 \%\) will fall between 65.5 and 71.8 beats per minute. (g) Of random samples of this size taken from students at this college, \(95 \%\) will have mean pulse rates between 65.5 and 71.8 beats per minute.

Short answer

Each statement in the exercise is a common misinterpretation of the confidence interval due to misunderstanding of the definition and meaning of a confidence interval. The only statement that correctly understands the meaning of a 95% confidence interval in this context is statement (g).

Step-by-step solution

Analyzing Statement (a)

Statement (a) is incorrect because a confidence interval does not guarantee that all individuals in a population will have values within that interval. The confidence interval pertains to the estimation of the population mean, not individual measurements.

Analyzing Statement (b)

Statement (b) is incorrect because the mean pulse rate for the sample has already been calculated, and is the point estimate used to calculate the confidence interval. The confidence interval is for the population mean, not the sample mean.

Analyzing Statement (c)

Statement (c) is correct in stating that the interval pertains to a population mean (in this case, the mean pulse rate for all students at the college), but is incorrect in asserting a certainty of 95%. Confidence levels represent a frequency for a class of hypothetical replications, not a probability for a single fixed interval.

Analyzing Statement (d)

Statement (d) is incorrect because the confidence interval does not predict what percentage of the student’s pulse rates fall within that interval. Instead, it indicates where the true population mean is estimated to be, with a certain level of confidence.

Analyzing Statement (e)

Statement (e) is incorrect because the sample was only taken from her college. The interval cannot be used to make statements about the mean pulse rate for all US college studentss.

Analyzing Statement (f)

Statement (f) is incorrect. A confidence interval does not make statements about individual means falling within the interval, it estimates where the true population mean falls.

Analyzing Statement (g)

Statement (g) is correct. It appropriately interprets the concept of confidence intervals, saying that if we were to repeatedly draw samples and calculate confidence intervals, we would expect \(95 \%\) of those intervals to contain the population mean pulse rate.

Key concepts

Misinterpretation of Confidence Intervals

Confidence intervals are a cornerstone of statistical analysis, yet they are often misconstrued, leading to misleading conclusions. A confidence interval gives an estimated range that is likely to contain the population mean, considering a specified confidence level, such as 95%. However, it's crucial to note that the true population parameter is fixed, and it’s our estimate that is subject to variability.

When a student claims to be '95% sure' that the mean or individual values fall within the confidence interval, they are walking into a common trap. Such statements suggest that the interval has a 95% probability of holding the true mean or individual values, which is incorrect. The '95%' refers to the method used to create the interval, indicating that if we took many samples and built intervals in the same manner, approximately 95% of them would capture the true population mean.

Population Mean Estimation

The purpose of estimating the population mean is to infer about a whole population from a sample statistic. The confidence interval uses the sample data to estimate where the true population mean lies. It's essential to recognize that the interval itself is based on the sample mean and the variability of the data, along with a margin of error. This margin reflects the sampling variability and the confidence level we've chosen.

For instance, stating the mean pulse rate for a sample of students will fall within a specific interval misinterprets the purpose, as the sample mean is a single point estimate. The interval estimates where the mean for all students, the population mean, is expected to lie. Obtaining a precise value for the population mean from a sample is not possible; hence, the interval provides a range for estimation.

Statistical Confidence Levels

Statistical confidence levels indicate how confident we can be in the methods used to estimate the population parameter. A '95% confidence level' does not mean there is a 95% probability of a specific outcome. Instead, it reflects the long-term frequency of confidence intervals that will contain the true population mean if the same study were repeated numerous times.

It's like a net catching fish – a '95% confidence net' won't be sure to catch any specific fish, but in the long run, it's constructed to catch the true 'fish' around 95 times out of 100. Misinterpreting this as a one-time probability overlooks how confidence levels actually function in statistics.

Sampling and Statistical Inference

Sampling is the process of selecting a subset from a larger population to estimate characteristics of the whole. Statistical inference then allows us to draw conclusions about that population. This is where the concept of a 'random sample' is fundamental. It ensures that every individual in the population has an equal chance of being included, which helps in generalizing the results to the broader population.

However, it is essential to clarify that even a well-conducted sample can have limitations in inference. For example, the interval estimated from one college's students cannot be generalized to all US college students, as the particular sample may not represent the larger student body adequately. Assumptions, variability, and potential biases must be considered when extending findings from a sample to a larger group.