Question

A stringer of tennis rackets has found that the actual string tension achieved for any individual racket stringing will vary as much as 6 pounds per square inch from the desired tension set on the stringing machine. If the stringer wishes to string at a tension lower than that specified by a customer only \(5 \%\) of the time, how much above or below the customer's specified tension should the stringer set the stringing machine? (NOTE: Assume that the distribution of string tensions produced by the stringing machine is normally distributed, with a mean equal to the tension set on the machine and a standard deviation equal to 2 pounds per square inch.)

Short answer

Answer: The stringer should set the stringing machine tension approximately 3.29 pounds per square inch below the customer's specified tension.

Step-by-step solution

Identify the given information

We are given the following information: 1. String tension is normally distributed. 2. Mean (μ) is equal to the tension set on the machine. 3. Standard deviation (σ) is 2 pounds per square inch. 4. We need to find the value of tension such that the machine produces tension lower than specified 5% of the time.

Calculate the z-score corresponding to 5% in the lower tail

To find the tension for the stringing machine, we will first calculate the z-score corresponding to the 5% in the lower tail of the normal distribution. Using a z-table, we can determine the z-score value: z = -1.645 (approximately)

Use the z-score formula to find the tension

Now, we'll use the z-score formula to determine the tension for the stringing machine: z = (X - μ) / σ Where z is the z-score, X is the desired tension for the stringing machine, μ is the customer's specified tension, and σ is the standard deviation (2 pounds per square inch). Solving for X: X = μ + (z * σ)

Substitute the z-score and standard deviation into the formula

Now, we'll plug in the values of z and σ into the formula: X = μ + (-1.645 * 2) X = μ - 3.29

Interpret the result

The stringer should set the stringing machine tension at approximately 3.29 pounds per square inch below the customer's specified tension. This will result in a tension lower than the specified tension 5% of the time.

Key concepts

Z-Score Calculation

Understanding the concept of a z-score is crucial when dealing with normal distribution in probability and statistics. A z-score, simply put, indicates how many standard deviations an element is from the mean. In the context of the exercise with the tennis rackets, the z-score calculation helps the stringer determine how to set the tension to avoid exceeding the customer's specified tension more than 5% of the time.

Let's break down the z-score into an even simpler definition: if a z-score is 0, it means the data point's value is exactly the mean value. A positive z-score means the data point is above the mean, and a negative z-score means it's below. In our scenario, we were looking for a negative z-score because we want to find a tension that's less than the specified tension. We used a z-score of -1.645 because it corresponds to the 5th percentile of the normal distribution, meaning only 5% of outcomes would fall below this point.

Standard Deviation

The concept of standard deviation is pivotal in understanding the dispersion of a dataset. Standard deviation measures how spread out the numbers in a data set are around the mean. In the case of our tennis racket stringer, the set standard deviation is 2 pounds per square inch. This number is vital because it helps in understanding the variability of the string tension.

Why does this matter? Think of standard deviation as a ruler that tells us the scale of deviation from the mean. A small standard deviation indicates that the values tend to be close to the mean, while a larger one shows that the values are spread out over a wider range. Knowing this, the stringer can anticipate the variability of the tension and adjust accordingly. The stringer uses the standard deviation in combination with the z-score to calculate by how much to adjust the tension setting on the machine.

Probability and Statistics

Grasping the fundamental principles of probability and statistics is essential in everyday decision-making processes, especially when they involve an element of uncertainty – such as the tension of a tennis racket string. Probability enables us to quantify the uncertainty, providing a way to predict the likelihood of various outcomes.

In the tennis racket stringing example, statistics come into play when we assume the distribution of string tensions follows a normal distribution. This assumption lets us use the properties of the normal distribution – such as the empirical rule and z-scores – to calculate probabilities and make informed decisions. The probability of 5% is the guide for our stringer to set the tension so that there's only a small chance the actual tension will be less than what the customer desires. By integrating probability with statistics, the stringer can assure quality control and customer satisfaction.