91Ó°ÊÓ

A study of the ability of individuals to walk in a straight line ("Can We Really Walk Straight?" Amer. J. of Physical Anthro., 1992: 19-27) reported the accompanying data on cadence (strides per second) for a sample of \(n=20\) randomly selected healthy men. \(\begin{array}{rrrrrrrrrr}.95 & .85 & .92 & .95 & .93 & .86 & 1.00 & .92 & .85 & .81 \\ .78 & .93 & .93 & 1.05 & .93 & 1.06 & 1.06 & .96 & .81 & .96\end{array}\) A normal probability plot gives substantial support to the assumption that the population distribution of cadence is approximately normal. A descriptive summary of the data from MINITAB follows: \(\begin{array}{lccccc}\text { Variable } \mathbb{N} & \text { Mean } & \text { Median } & \text { TrMean } & \text { StDev } & \text { SEMean } \\ \text { cadence } 20 & 0.9255 & 0.9300 & 0.9261 & 0.0809 & 0.0181 \\ \text { Variable } & \text { Min } & \text { Max } & \text { Q1 } & \text { Q3 } & \\ \text { cadence } & 0.7800 & 1.0600 & 0.8525 & 0.9600 & \end{array}\) a. Calculate and interpret a \(95 \%\) confidence interval for population mean cadence. b. Calculate and interpret a \(95 \%\) prediction interval for the cadence of a single individual randomly selected from this population. c. Calculate an interval that includes at least \(99 \%\) of the cadences in the population distribution using a confidence level of \(95 \%\).

Step by step solution

01

Identify the Data Given

We are working with samples of men's cadence rates, with a sample size of \( n = 20 \). We know the mean cadence is 0.9255 and the standard deviation is 0.0809.

02

Calculate 95% Confidence Interval for Population Mean

For a 95% confidence interval, we need the t-distribution critical value for \( n-1 = 19 \) degrees of freedom. This is approximately \( t_{0.025, 19} = 2.093 \). The confidence interval is calculated as \[ \text{CI} = \bar{x} \pm t \cdot \frac{s}{\sqrt{n}} \] where \( \bar{x} = 0.9255 \), \( s = 0.0809 \), and \( n = 20 \). Thus, CI becomes: \[ 0.9255 \pm 2.093 \cdot \frac{0.0809}{\sqrt{20}} \approx 0.9255 \pm 0.0378 \] which means the interval is \( [0.8877, 0.9633] \).

03

Interpret the Confidence Interval

The 95% confidence interval for the mean cadence is \( [0.8877, 0.9633] \). This means we can be 95% confident that the true mean cadence of the population falls within this interval.

04

Calculate 95% Prediction Interval for a Single Individual

The formula for a prediction interval is \[ \bar{x} \pm t \cdot \sqrt{s^2 \left(1 + \frac{1}{n}\right)} \]. Using the values \( \bar{x} = 0.9255 \), \( t = 2.093 \), \( s = 0.0809 \), and \( n = 20 \), we calculate: \[ 0.9255 \pm 2.093 \cdot \sqrt{0.0809^2 \left(1 + \frac{1}{20}\right)} \approx 0.9255 \pm 0.1869 \] which gives us the interval \( [0.7386, 1.1124] \).

05

Interpret the Prediction Interval

The 95% prediction interval for the cadence of a single individual is \( [0.7386, 1.1124] \). This means we are 95% confident that a single randomly selected individual's cadence will fall within this interval.

06

Calculate Interval for 99% of Population Cadences

To include at least 99% of the population, we use the formula for an interval based on the normal distribution: \( \bar{x} \pm z \cdot s \), where \( z = 2.576 \) for a 99% interval. Thus, \[ 0.9255 \pm 2.576 \cdot 0.0809 \approx 0.9255 \pm 0.2084 \] giving us the interval \( [0.7171, 1.1339] \).

07

Interpret the Inclusion Interval for Population Cadences

The interval \( [0.7171, 1.1339] \) means at least 99% of the population cadences will fall within this range. This is calculated at a 95% confidence level.

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Key Concepts

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

Confidence Interval

A confidence interval is a range of values that is used to estimate the true value of a population parameter, such as the mean. In this exercise, we calculated a 95% confidence interval for the average cadence (strides per second) of a group of healthy men. This interval gives us an estimate of where the true population mean might lie.

To construct a confidence interval, we use the sample mean, sample standard deviation, and a critical value from the t-distribution, since we have a small sample size of 20. The formula used is: \[ \text{CI} = \bar{x} \pm t \cdot \frac{s}{\sqrt{n}} \]where:

• \( \bar{x} \) is the sample mean, here 0.9255
• \( t \) is the critical value from the t-distribution with \( n-1 = 19 \) degrees of freedom (approximately 2.093 for a 95% confidence level)
• \( s \) is the sample standard deviation, 0.0809
• \( n \) is the sample size, 20

The result, \([0.8877, 0.9633]\), tells us that we can be 95% confident that the true mean cadence of the population falls within this interval. This does not mean that 95% of the data lies within this range, but rather that if we were to take many samples and build an interval each time, 95% of those intervals would contain the true mean.

Prediction Interval

A prediction interval is different from a confidence interval, as it is used to predict a single future observation from the population. In our exercise, we calculated a 95% prediction interval for the cadence of one randomly selected individual. This interval estimates the range in which we expect the cadence of one new person to fall.

The prediction interval is calculated using the formula:\[ \bar{x} \pm t \cdot \sqrt{s^2 \left(1 + \frac{1}{n}\right)} \]where all variables are as defined earlier but note that it accounts for the variability both within samples and between future samples.
The calculated prediction interval for the individual's cadence is \([0.7386, 1.1124]\). This means that we expect the cadence of a single randomly selected healthy man from this population to fall within this interval 95% of the time. It is wider than a confidence interval due to the additional uncertainty associated with predicting a single observation.

Population Distribution

The population distribution refers to how values of a population are spread or distributed. In this exercise, we are assuming that the cadence of the population is normally distributed. This assumption is key in making valid inferences about the population from the sample data.

A normal distribution is a bell-shaped curve and is characterized by its mean and standard deviation. Knowing the distribution helps us apply statistical methods such as calculating confidence and prediction intervals. A normal probability plot has been used to verify this normal distribution assumption. The plot supported the idea that the cadence data closely follows a normal distribution.
When we assume a normal distribution, we can also calculate an interval to include nearly all population values. For example, using the normal distribution properties, we calculated a range that includes at least 99% of the population cadences using a confidence level of 95%. This interval, \([0.7171, 1.1339]\), captures the breadth of cadences expected in nearly all individuals in the population, accounting for the variation both within and across larger data sets.