Question

Smoking and Blood Pressure An experiment was conducted to estimate the effect of smoking on the blood pressure of a group of 35 cigarette smokers. The difference for each participant was obtained by taking the difference in the blood pressure readings at the beginning of the experiment and again five years later. The sample mean increase, measured in millimeters of mercury, was \(\bar{x}=9.7\). The sample standard deviation was \(s=5.8\). Estimate the mean increase in blood pressure that one would expect for cigarette smokers over the time span indicated by the experiment. Find the margin of error. Describe the population associated with the mean that you have estimated.

Short answer

Answer: The estimated mean increase in blood pressure for cigarette smokers over a five-year period is approximately 9.7 mmHg. The margin of error for this estimate is approximately 1.918 mmHg.

Step-by-step solution

Identify the sample mean, sample standard deviation, and sample size

For this problem, we're given the following information about the sample: Sample mean increase in blood pressure (\(\bar{x}\)): \(9.7\) mmHg Sample standard deviation (\(s\)): \(5.8\) mmHg Sample size (\(n\)): \(35\)

Determine the level of confidence, and find the z-score

We're going to use a 95% confidence level for estimating the mean increase in blood pressure. From the standard normal distribution table or using a calculator, we can find the corresponding z-score (\(z\)): \(z = 1.96\)

Calculate the standard error

The standard error is calculated using the sample standard deviation divided by the square root of the sample size: Standard error (\(SE\)) = \(\frac{s}{\sqrt{n}}\) \(SE = \frac{5.8}{\sqrt{35}} \approx 0.979\)

Calculate the margin of error

The margin of error is calculated using the z-score and the standard error: Margin of error (\(ME\)) = \(z \times SE\) \(ME = 1.96 \times 0.979 \approx 1.918\)

Construct the 95% confidence interval

The 95% confidence interval is calculated by adding and subtracting the margin of error from the sample mean: Lower limit = \(\bar{x} - ME \approx 9.7 - 1.918 \approx 7.782\) Upper limit = \(\bar{x} + ME \approx 9.7 + 1.918 \approx 11.618\) Therefore, the 95% confidence interval for the mean increase in blood pressure for cigarette smokers over the five-year span is approximately between \(7.782\) mmHg and \(11.618\) mmHg.

Describe the population

The population associated with the estimated mean is the entire group of cigarette smokers. The mean increase in blood pressure of \(9.7\) mmHg is an estimate of the average change in blood pressure for all cigarette smokers over a five-year period. In conclusion, we can say with 95% confidence that, on average, cigarette smokers could experience an increase in blood pressure between \(7.782\) mmHg and \(11.618\) mmHg over five years. The margin of error for this estimate is approximately \(1.918\) mmHg.

Key concepts

Sample Mean

In statistics, the sample mean is a critical component when analyzing data collected from a subset of a larger population. The sample mean represents the average of all measurements or values in the sample. For the given problem, the sample mean increase in blood pressure \( \bar{x} = 9.7 \) mmHg indicates the average rise in blood pressure over five years for the 35 cigarette smokers participating in the study. This sample mean acts as an estimator of the population mean, helping us infer properties about the broader group of all cigarette smokers.

• It helps in understanding the central tendency of your data.
• It acts as a starting point for constructing confidence intervals.

Using this value, we can estimate the overall behavior or trend in the larger population based on the smaller group.

Standard Deviation

The standard deviation is a measure of how much variation or dispersion exists from the sample mean. This calculation gives us insight into how spread out the data points are in our sample. In this problem, the standard deviation of the increase in blood pressure for the sample is given as \( s = 5.8 \) mmHg.
This means that, on average, each individual's blood pressure increase is about 5.8 mmHg away from the sample mean of 9.7 mmHg both above and below.

• It helps identify data volatility.
• A lower standard deviation indicates that the data points tend to be closer to the mean.
• A higher standard deviation indicates a wider spread around the mean.

It is a useful tool for determining how representative your sample mean is for a larger population, helping inform the reliability of the sample mean as an estimator of the population mean.

Margin of Error

The margin of error gives us an indication of the potential error in our estimates in the context of confidence intervals. It represents the range of values that the true population parameter is expected to fall within with a given level of confidence. For this exercise, the calculated margin of error is \( ME \approx 1.918 \) mmHg. The formula used is:\[ ME = z \times \frac{s}{\sqrt{n}} \]Where:

• \( z \) is the z-score corresponding to the confidence level (1.96 for 95% confidence).
• \( s \) is the sample standard deviation.
• \( n \) is the sample size.

This metric plays a crucial role in quantifying the extent of uncertainty associated with sample statistic estimates.

Population Description

Understanding the population description is essential as it refers to the group from which the sample is drawn and to which the results are generalized. In this context, the population of interest is all cigarette smokers, a broader group beyond just the 35 participants in the study. The mean increase in blood pressure of the sample provides an estimate of the average change we might observe in blood pressure for all cigarette smokers over the five years. Keeping in mind the limitations of sample size and variability, this description helps in making broader conclusions about the likely health impacts of smoking across a more extensive population.

• The sample data acts as a tool to make educated guesses about the larger population.
• It helps in specifying the nature of the broader population whose characteristics we aim to understand through our sample analysis.

By carefully describing and understanding the population, researchers can better communicate their findings and suggest interventions or policy changes related to public health.