91Ó°ÊÓ

You have measured the systolic blood pressure of an SRS of 25 company employees. A \(95 \%\) confidence interval for the mean systolic blood pressure for the employees of this company is \((122,138) .\) Which of the following statements is true? (a) \(95 \%\) of the sample of employees have a systolic blood pressure between 122 and 138 . (b) \(95 \%\) of the population of employees have a systolic blood pressure between 122 and 138 . (c) If the procedure were repeated many times, \(95 \%\) of the resulting confidence intervals would contain the population mean systolic blood pressure. (d) If the procedure were repeated many times, \(95 \%\) of the time the population mean systolic blood pressure would be between 122 and 138 . (e) If the procedure were repeated many times, \(95 \%\) of the time the sample mean systolic blood pressure would be between 122 and 138 .

Step by step solution

01

Understanding the Confidence Interval

A 95% confidence interval means that if we were to take many samples and compute a confidence interval for each, about 95% of these intervals would contain the true population mean.

02

Analyze Option (a)

Option (a) suggests that 95% of the sample have blood pressures between 122 and 138. The confidence interval applies to the population mean, not individual sample values, making this statement incorrect.

03

Analyze Option (b)

Option (b) suggests that 95% of the population has blood pressures between 122 and 138. Similarly, a confidence interval estimates the range for the population mean, not individual data points, so this statement is also incorrect.

04

Analyze Option (c)

Option (c) is a correct interpretation of the confidence interval. If the process of obtaining confidence intervals were repeated many times, 95% of those intervals would contain the true mean systolic blood pressure of the population.

05

Analyze Option (d)

Option (d) incorrectly suggests that the true population mean itself varies between 122 and 138 in repeated testing. The population mean is fixed; the intervals vary.

06

Analyze Option (e)

Option (e) misrepresents the concept by applying it to sample means. The interval estimates the range for the population mean, not where sample means would fall.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Population Mean

The population mean represents the average value of a specific measurement for an entire population. In statistical terms, it is denoted by the symbol \( \mu \). In the context of systolic blood pressure, if we were to measure the blood pressure of every single employee in a company, the calculated average would be the population mean.

However, since it is often impractical to measure every individual in a large population, we rely on samples to estimate this mean. The confidence interval, such as the one between 122 and 138 in this exercise, provides a range within which we are 95% confident the true population mean of the systolic blood pressure lies. Remember, the goal of constructing confidence intervals is to estimate the population mean, not each individual's measurements.

• Population mean is denoted as \( \mu \).
• Represents the average of all members in a population.
• Confidence intervals help estimate the population mean.

Systolic Blood Pressure

Systolic blood pressure is the force your blood exerts against the walls of your arteries when your heart beats. It's the top number in a blood pressure reading, such as 120/80 mmHg. Regular monitoring of systolic blood pressure is crucial as it helps in assessing the risk for heart disease and stroke.

In this exercise, the systolic blood pressure readings of 25 employees were used to construct a confidence interval. These readings give us a snapshot, and the confidence interval helps infer what the typical systolic blood pressure could be for the entire population of employees.

• Indicator of heart and artery health.
• Key factor in calculating health risks.
• Part of the data used to estimate the population mean.

Sampling Methods

Sampling methods are techniques used to select a subset of individuals from a population to estimate the population parameters, like the mean. In this exercise, a Simple Random Sample (SRS) was used.

A simple random sample ensures that every member of the population has an equal chance of being selected. This method helps to avoid bias and achieve a representative sample. Thus, when calculating the 95% confidence interval for systolic blood pressure, we ensure that the range will likely include the true population mean if repeated sampling is done.

• Prevents bias in data collection.
• Provides a fair estimation of population parameters.
• Essential for reliable confidence intervals.