Question

Find endpoint(s) on a \(N(0,1)\) density with the given property. ( a) The area to the left of the endpoint is about 0.70 (b) The area to the right of the endpoint is about \(0.01 .\) (c) The area between \(\pm z\) is about 0.90 .

Short answer

Based on the z-values found from the Z Table, the endpoints for part (a) is approximately 0.52, for part (b) is approximately 2.33, and for part (c) are approximately -1.645 and 1.645.

Step-by-step solution

For Part (a)

We need to find the z-score such that the area to its left under the standard normal curve is approximately 0.7. We can use a standard normal distribution table (Z Table) to find the z-score that corresponds closest to 0.7. From the table, when area to the left of z is 0.7, z is about 0.52.

For Part (b)

This part asks for the z-score such that the area to the right of it under the standard normal curve is approximately 0.01. To obtain this, we need to first find the z-score that corresponds to an area to the left of 0.99 (because 1 - 0.01 = 0.99) from the Z Table. Looking up this value from the table, the z-score is about 2.33.

For Part (c)

This part requires determining the z-score such that the area between -z and +z under the standard normal curve is approximately 0.9. Because the standard normal distribution is symmetrical around 0, the cumulative area from -z to +z of 0.9 implies that the cumulative area from -z to 0 and from 0 to +z are each 0.45. Referring to the Z Table, a cumulative area under curve of 0.45 from 0 is closest to a z-score of 1.645. Hence, z = -1.645 and z = 1.645 are our endpoints.

Key concepts

Understanding the Standard Normal Distribution

The standard normal distribution is a crucial concept in statistics, representing a continuous probability distribution that is symmetrical and bell-shaped, known as the 'normal curve'. At its core, it is a distribution that has a mean of zero (\( \bar{x} = 0 \) and a standard deviation of one (\( \bar{\text{\textsigma}} = 1 \) which makes it a special case of the normal distribution.

When plotting the data that follows this distribution, we observe that most values cluster around the mean and fall off evenly as we move away to the left or right, creating the characteristic bell shape. It's important because this pattern occurs frequently in natural and social phenomena, from test scores to heights. An essential feature of the standard normal distribution is its standardization. By converting scores from any normal distribution to z-scores, a common scale is established, allowing for comparison between different data sets.

For a given value 'x' in a standard normal distribution, the z-score is calculated using the formula \( z = \frac{x - \bar{x}}{\bar{\text{\textsigma}}} \) where \( \bar{x} \) is the mean and \( \bar{\text{\textsigma}} \) is the standard deviation of the distribution. This z-score tells us how many standard deviations an element is from the mean, and it is this scaling that allows the use of the z-score to compare different sets of data.

Exploring Normal Density

Normal density refers to the function that describes the probability density of a normal distribution. It is represented by the equation \( f(x) = \frac{1}{\bar{\text{\textsigma}}\sqrt{2\pi}} e^{-\frac{1}{2}(\frac{x - \bar{x}}{\bar{\text{\textsigma}}})^2} \) where \( f(x) \) is the probability density function, and \( e \) is the base of the natural logarithm.

The 'height' of the curve at any given point x corresponds to the value of \( f(x) \) and gives the density of the distribution at that point. In a standard normal distribution, the density function simplifies to \( f(z) = \frac{1}{\text{\textsqrt}{2\pi}} e^{-\frac{1}{2}z^2} \) since the mean (\( \bar{x} \) is 0 and standard deviation (\( \bar{\text{\textsigma}} \) is 1.

The curve's shape is determined by its mean and standard deviation. The total area under the curve equals 1, signifying the entire probability space. This probability of finding a value within a certain range can be calculated by finding the area under the curve, which is where the concept of z-scores and the Z Table comes into play.

Navigating the Z Table

The Z Table, also known as the 'Standard Normal Table', is a reference table that provides the probability of a standard normal variable falling within a particular range. It typically shows the probability that a standard normal random variable is less than or equal to a particular value.

Usually, Z Tables are arranged such that the body of the table lists z-scores to two decimal places along the vertical column, representing z-values, and provides cumulative probabilities or areas to the left of those z-values in the body. Some tables include the area to the right instead or provide both.

When using a Z Table, it's important to know the direction of the inequality – whether you are looking for the area to the left (less than) or to the right (greater than) of a z-score. For example, to find the endpoint in a problem where the area to the left of the endpoint is about 0.70, you'd look for 0.70 in the table and find the corresponding z-score.

To interpret the Z Table effectively, remember that it only covers the area for the standard normal distribution. For non-standard normal distributions, it's necessary to first convert the x-values into z-scores before consulting the table. Accuracy in reading and interpolation skills are important when using this table, as the exact value you're looking for may not be listed, and you may need to estimate between two given values.