Question

A distribution of measurements has a mean of 75 and a standard deviation of \(5 .\) You know nothing else about the size or shape of the data. Use this information to find the proportion of measurements in the intervals given. Between 60 and 90

Short answer

Answer: Approximately 99.74% of the measurements fall between 60 and 90.

Step-by-step solution

Calculate the z-scores for interval values

To compute the z-score for the interval values (60 and 90), you need to subtract the mean (75) and divide by the standard deviation (5). For 60: \( z_{60} = \frac{60 - 75}{5} = -3 \) For 90: \( z_{90} = \frac{90 - 75}{5} = 3 \)

Find the probability using the z-table

Now, you'll use the z-table to find the probability of the given z-scores. The z-table gives the probabilities to the left of a z-score in a standard normal distribution. For \(z_{60} = -3\), corresponding probability = 0.0013 For \(z_{90} = 3\), corresponding probability = 0.9987

Find the proportion of measurements between 60 and 90

Subtract the probability of the lower z-score from the probability of the higher z-score to find the proportion of measurements in the given interval. Proportion = \(P(z_{90}) - P(z_{60}) = 0.9987 - 0.0013 = 0.9974\) Therefore, approximately 99.74% of the measurements fall between 60 and 90.

Key concepts

Standard Deviation

Understanding standard deviation is crucial when dealing with statistical data. It is a measure that indicates how much the values of a dataset deviate from the mean (average) of the dataset. This concept is foundational in statistics because it provides insight into the variability or spread of the data points.

For instance, if you encounter a set of test scores with a low standard deviation, it implies that most of the scores are clustered close to the mean, suggesting consistency among the test takers. Conversely, a high standard deviation would point to scores that are spread out over a wider range, indicating a greater disparity in performance. In the exercise, a standard deviation of 5 implies that the majority of measurements fall within 5 units of the mean on either side, creating a relatively narrow spread if we're discussing something like test scores.

Calculating standard deviation involves several steps, including finding the mean, determining the variance by averaging the squared differences from the mean, and finally, taking the square root of this variance. But in the context of the exercise we're discussing, it's given as part of the problem, simplifying our process considerably.

Normal Distribution

The concept of normal distribution, also known as the Gaussian distribution, is pivotal in statistics. It's often represented by a symmetrical, bell-shaped curve where the mean, median, and mode of the distribution all coincide at the peak of the curve. One of the remarkable properties of a normal distribution is that it's completely determined by its mean and standard deviation.

The curve of the normal distribution has points of inflection; these are one standard deviation away from the mean. Approximately 68% of the data in a normal distribution is within one standard deviation of the mean, 95% is within two standard deviations, and about 99.7% within three standard deviations. These percentages are a foundation of what is known as the empirical rule or 68-95-99.7 rule.

In our exercise example, because the distribution is described only by the mean and standard deviation, it suggests that we're assuming a normal distribution of measurements. This assumption is quite common in many real-world scenarios due to the central limit theorem, which states that the means of sufficiently large samples from a population will be approximately normally distributed, regardless of the population's distribution shape.

Z-table

The z-table is a powerful tool in statistics for understanding the probabilities associated with a normal distribution. It is essentially a reference chart that provides the area (probability) to the left of a particular z-score on a standard normal distribution curve. A z-score itself is a statistical measure that describes a value's position relative to the mean of a group of values, measured in terms of standard deviations from the mean.

Using the z-table, you can find the probability of random variable values falling within a certain interval. For example, a z-score of 0 corresponds to the mean of the distribution, and the table will show a probability of 0.5 or 50%, meaning there's a 50-50 chance of a value falling below the mean. If you have a positive z-score, the table provides the area under the curve to the left of that z-score, indicating how likely it is for a value to be below a certain threshold.

In the context of our original exercise, the z-scores for 60 and 90 are calculated as -3 and 3, respectively. These scores are then used with the z-table to determine the proportion of measurements that fall between these values. It's key to grasp that the z-table enables us to interpret the standard scores (-3 and 3) into actual probabilities (0.0013 and 0.9987), achieving the final solution for the exercise.