Question

The mean and standard deviation of the speeds of the sample of 69 skaters at the end of the 6 -meter distance in Exercise 8.106 were 5.753 and .892 meters per second, respectively. a. Find a \(95 \%\) confidence interval for the mean velocity at the 6 -meter mark. Interpret the interval. b. Suppose you wanted to repeat the experiment and you wanted to estimate this mean velocity correct to within .1 second, with probability .99. How many skaters would have to be included in your sample?

Short answer

Answer: The 95% confidence interval for the mean velocity of skaters at the 6-meter mark is (5.602 m/s, 5.904 m/s). To estimate the mean velocity within 0.1 seconds with a 99% probability, 679 skaters need to be included in the sample.

Step-by-step solution

Identify the given parameters

We are given the sample mean (\(\bar{X}\)), standard deviation (\(\sigma\)), and sample size (\(n\)): \(\bar{X} = 5.753\) \(\sigma = 0.892\) \(n = 69\)

Determine the z-score for a 95% confidence interval

Using a z-table or statistical calculator, we find the corresponding z-score for the 95% confidence interval: \(z = 1.96\)

Apply the confidence interval formula

Using the provided information, we can now calculate the confidence interval: \(CI = 5.753 \pm 1.96\frac{0.892}{\sqrt{69}}\)

Calculate the confidence interval bounds

Lower bound: \(5.753 - 1.96\frac{0.892}{\sqrt{69}} = 5.602\) Upper bound: \(5.753 + 1.96\frac{0.892}{\sqrt{69}} = 5.904\) The 95% confidence interval for the mean velocity at the 6-meter mark is (5.602 m/s, 5.904 m/s). This means that we are 95% confident that the true mean velocity of skaters at the 6-meter mark lies between 5.602 and 5.904 meters per second. #b. Sample Size for Desired Precision#

Determine the z-score for a 99% confidence interval

Using a z-table or statistical calculator, we find the corresponding z-score for the 99% confidence interval: \(z = 2.576\)

Identify the margin of error and standard deviation

The margin of error (E) is given as 0.1 seconds, and the standard deviation is 0.892 meters per second.

Apply the sample size formula

Using the provided information, we can now calculate the required sample size for the desired precision: \(n = \left(\frac{2.576 \times 0.892}{0.1}\right)^2\)

Calculate the required sample size

\(n = (\frac{2.576 \times 0.892}{0.1})^2 \approx 678.46\) Since we cannot have a decimal value for the sample size, we round up to the nearest whole number: Required sample size: \(n = 679\) To estimate the mean velocity within 0.1 seconds with a 99% probability, 679 skaters need to be included in the sample.

Key concepts

Understanding Mean Velocity

Mean velocity is the average speed of a set of observations, often crucial in statistical analysis for evaluating performance metrics. In our context, it refers to the average speed of skaters measured at the end of a 6-meter distance. Calculating the mean velocity involves summing up the individual velocities of each skater and then dividing by the total number of skaters.

Mathematically, mean velocity \( \bar{X} \) is given by the formula:

• \( \bar{X} = \frac{\text{Sum of velocities}}{n} \)

where \( n \) is the number of observations or skaters in this scenario.

Understanding mean velocity helps in predicting trends and assessing performance consistency over the measured distance. It also forms the basis for further statistical calculations like confidence intervals.

Sample Size Calculation

Calculating the appropriate sample size is crucial for ensuring statistical confidence and accuracy in experimental results. For estimating mean velocity to a specific precision—such as within 0.1 seconds with 99% confidence—it’s essential to determine how many observations or skaters are needed.

The formula used for sample size calculation is:

• \( n = \left( \frac{Z \times \sigma}{E} \right)^2 \)

where:

• \( Z \) is the z-score corresponding to the desired level of confidence,
• \( \sigma \) is the standard deviation, and
• \( E \) is the margin of error acceptable in the calculation.

In practice, this calculation helps researchers know how large a sample is needed to estimate the average accurately. The larger the sample size, the more reliable the results, reducing the margin of error.

Decoding the Z-Score

The z-score is a statistical measurement that describes a value's relation to the mean of a group of values. It indicates how many standard deviations an element is from the mean. In the context of calculating confidence intervals, the z-score helps determine the range in which the true population parameter lies, based on the sample statistic.

For a 95% confidence level, the z-score is 1.96, meaning that the interval created will capture the true mean 95% of the time. Similarly, for 99% confidence, a z-score of 2.576 is used, providing a more precise estimate with a smaller chance of error.

The choice of z-score directly impacts the width of the confidence interval and is crucial for accurately interpreting statistical data.

Role of Standard Deviation

Standard deviation measures the amount of variation or dispersion of a set of values. A low standard deviation indicates that values tend to be close to the mean, while a high standard deviation suggests a wide range of values.

In this exercise, the standard deviation of 0.892 meters per second illustrates how much individual skaters’ velocities vary around the mean velocity of 5.753 m/s.

Standard deviation is crucial in calculating confidence intervals and required sample sizes because it impacts the width of those intervals. It helps in understanding the reliability and spread of the data, providing insights into the consistency and predictability of the observed performance.