Question

Find \(z_{0}\) such that \(P\left(-z_{0}

Short answer

Answer: The z-scores corresponding to a probability of 0.5 between them are approximately -0.67 and 0.67. They represent the 25th and 75th percentiles, respectively.

Step-by-step solution

Understand the problem

We are given a standard normal distribution and the probability that \(P(-z_0 < z < z_0) = 0.5\). We need to find the value of \(z_0\) and then the corresponding percentiles for \(-z_0\) and \(z_0\).

Identify limits of the integral and the area under the curve

Since the total area under the curve is 1 in a standard normal distribution, and we are looking for the probability within the range \((-z_0, z_0)\), we can divide the remaining area by 2 to find the probability on each tail. So, the area in each tail will be \((1 - 0.5) / 2 = 0.25\).

Find the z-score for the required area

We know that the standard normal distribution table shows the area to the left of a z-score. Since we know the area in the left tail, we can add this to the probability we are looking for: \(0.25 + 0.5 = 0.75\). Now, we need to find the z-score in the table that corresponds to an area of 0.75.

Obtain z-score from the z-table

Look for the closest value to 0.75 in the z-table. In this case, we find that a z-score of approximately \(z_0=0.67\) corresponds to an area of 0.75 to the left. So, \(z_0 \approx 0.67\).

Determine the percentiles

To find the percentiles, we need to calculate the cumulative probability up to the given z-scores. For \(-z_0\), we have \(P(Z<-z_0) = 0.25\), which represents the 25th percentile. For \(z_0\), we have \(P(Z<z_0) = 0.75\), which represents the 75th percentile. So, the z-scores \(-z_0\) and \(z_0\) represent the 25th and 75th percentiles, respectively.

Key concepts

Understanding Z-Scores

In the realm of statistics, the z-score is pivotal when discussing the standard normal distribution. It is a measure that tells us how many standard deviations a data point lies away from the mean.
This is extremely helpful because it allows us to compare different data points within a dataset or even between different datasets if normalization is achieved.
For instance, in understanding the problem at hand, we need to find the z-score that encapsulates the middle 50% of data in a standard normal distribution.

• A z-score of 0 denotes that the data point is exactly at the mean.
• Positive z-scores indicate data points above the mean, whereas negative z-scores point to those below the mean.

In practical terms, using a z-table, when asked to find a z-score that encompasses both -0.67 and 0.67, we explore the corresponding probabilities or areas, showing how data points relate to normal distribution.

Decoding Percentiles

Percentiles are a great way to compare scores across different datasets. They give us a sense of how a particular score relates to all other scores in a dataset.
They are quantiles which divide data into 100 equal parts, and the percentile rank signifies the percentage of scores in its frequency distribution that are equal to or lower than it.
In the context of the exercise, by understanding that the middle 50% of a dataset translates into the percentiles defined by -0.67 and 0.67 z-scores, we focus on key takeaways:

• The 25th percentile (-0.67) indicates the point below which approximately 25% of data falls.
• The 75th percentile (0.67) signifies a point below which approximately 75% of the data lies.

By aligning z-scores with percentiles, students can comprehend how specific data values fit into the broader distribution picture.

Interpreting Probability with Normal Distribution

Probability in a normal distribution reflects how likely it is for a particular outcome or range of outcomes to occur.
The standard normal distribution, a normal distribution with a mean of 0 and a standard deviation of 1, is often used as a benchmark. When we talk about probabilities, such as the area between -0.67 and 0.67 already accounting for 0.5 of the total probability, these are effectively percentages that describe certainty.

• The probability of 0.5 indicates that there is a 50% chance data falls within a given range (-0.67 to 0.67).
• Tails of the distribution, each holding a probability of 0.25, illustrate the likelihood of data lying outside the middle 50%.

By grasping how probability relates to areas under the curve in a normal distribution, students see how data points spread over different ranges and what that implies for analyses.