Question

Two random samples of 32 individuals were selected. One sample participated in an activity which simulates hard work. The average breath rate of these individuals was 21 breaths per minute. The other sample did some normal walking. The mean breath rate of these individuals was \(14 .\) Find the \(90 \%\) confidence interval of the difference in the breath rates if the population standard deviation was 4.2 for breath rate per minute.

Short answer

The 90% confidence interval for the difference in breath rates is (4.56, 9.44).

Step-by-step solution

Understanding the Confidence Interval Formula

To find the confidence interval, we'll use the formula for the confidence interval of the difference between two means: \( \text{CI} = (\bar{x}_1 - \bar{x}_2) \pm Z \cdot \sqrt{\frac{\sigma^2}{n_1} + \frac{\sigma^2}{n_2}} \), where \( \bar{x}_1 \) and \( \bar{x}_2 \) are the means of the two samples, \( \sigma \) is the population standard deviation, and \( n_1 \) and \( n_2 \) are the sample sizes.

Finding the Z-value for 90% Confidence Level

The Z-value for a 90% confidence interval is approximately 1.645, assuming a standard normal distribution. This value can be found using a Z-table or calculator for the critical value corresponding to a 90% confidence level.

Applying the Values into the Formula

We have \( \bar{x}_1 = 21 \), \( \bar{x}_2 = 14 \), \( \sigma = 4.2 \), and both \( n_1 = n_2 = 32 \). Substitute these values into the confidence interval formula: \[ \text{CI} = (21 - 14) \pm 1.645 \times \sqrt{\frac{4.2^2}{32} + \frac{4.2^2}{32}} \] This simplifies to \[ \text{CI} = 7 \pm 1.645 \times \sqrt{\frac{17.64}{32} + \frac{17.64}{32}} \]

Calculating the Standard Error

Calculate the standard error part of the formula:\[ \text{Standard Error} = \sqrt{\frac{17.64}{32} + \frac{17.64}{32}} = \sqrt{1.1025 + 1.1025} = \sqrt{2.205} \approx 1.485 \]

Calculation of Confidence Interval Range

Now multiply the standard error by the Z-value:\[ 1.645 \times 1.485 \approx 2.44 \] Add and subtract this value from the difference in means (7) to find the confidence interval:\[ 7 \pm 2.44 \] This yields a confidence interval of \( (4.56, 9.44) \).

Conclusion

The 90% confidence interval for the difference in breath rates is approximately \((4.56, 9.44)\). This means we are 90% confident that the true difference in average breath rates between the two activities lies within this range.

Key concepts

Difference of Means

The "difference of means" refers to the comparison between the average values (means) of two different groups or samples.
This concept is especially useful when we want to measure how different two groups are from each other in terms of a particular characteristic.

In the given exercise, we have two groups based on breath rates: one group is engaged in a hard work simulation, and the other is involved in normal walking.
By calculating the difference of their average breath rates, we aim to analyze the impact of different activities on the breath rate.

To find the difference of means, we simply subtract the mean of the second group from the mean of the first group. Here, it was calculated as \(21 - 14 = 7\).
This difference of 7 breaths per minute serves as a measure to provide an initial overview of how distinct the two groups are regarding breath rate during their varying activities.

Standard Deviation

Standard deviation measures the amount of variation or dispersion of a set of values.
It indicates how much the individual data points differ from the mean of the dataset.
This statistical tool helps us understand data variability and consistency.

In our exercise, the population standard deviation, denoted as \(\sigma\), was given as 4.2.
This value tells us that the individual breath rates demonstrate a variation of approximately 4.2 breaths per minute from the average breath rate of the population.

The standard deviation is crucial because it plays a key role in calculating the confidence interval, impacting how narrow or broad the interval range will be.

Standard Error

The "standard error" is a statistical term that quantifies the amount by which the sample mean differs from a true population mean.
It gives us an idea of how close our sample mean is likely to be to the population mean.

To calculate the standard error of the differences between two means, we use the formula:
\[ \text{Standard Error} = \sqrt{\frac{\sigma^2}{n_1} + \frac{\sigma^2}{n_2}}\]
Where \(\sigma\) is the population standard deviation, and \(n_1\) and \(n_2\) are the sample sizes for each group.

In the context of our exercise, after inserting the provided values, the standard error was calculated to be approximately 1.485.
This value essentially tells us about the expected variability of the difference in sample means from the true difference in population means.

Z-value

A "Z-value" (or Z-score) relates to the standard deviation under a standard normal distribution.
It's a way of identifying the position of an element within the distribution in comparison to the mean.

When calculating a confidence interval, the Z-value helps determine how confident we can be in our interval coefficient's accuracy.
The Z-value is based on the desired confidence level. For a 90% confidence interval, we look for the Z-value that corresponds to an area of 0.90 in the cumulative standard normal distribution.

In the exercise, the Z-value was found to be approximately 1.645 for the 90% confidence level.
This value directly influences the margin of error in our confidence interval by being multiplied with the computed standard error.