In Exercises 2–4, the random variable x is normally distributed with mean mu= 18 and standard deviation sigma 7.6


Find the value of x that has 88.3% of the distribution’s area to its left.

Verified step by step guidance

1. Step 1: Understand the problem. We are tasked with finding the value of x in a normal distribution where the mean (μ) is 18, the standard deviation (σ) is 7.6, and 88.3% of the distribution's area lies to the left of x. This means we are looking for the x-value corresponding to a cumulative probability of 0.883.
2. Step 2: Convert the cumulative probability to a z-score. Use the standard normal distribution table or a statistical software to find the z-score (z) that corresponds to a cumulative probability of 0.883. The z-score represents the number of standard deviations x is away from the mean in a standard normal distribution.
3. Step 3: Use the z-score formula to solve for x. The formula is: $x=\mu +z\times \sigma$, where μ is the mean, z is the z-score, and σ is the standard deviation.
4. Step 4: Substitute the known values into the formula. Replace μ with 18, σ with 7.6, and z with the value obtained in Step 2. This will give you the value of x.
5. Step 5: Interpret the result. The calculated x-value represents the point on the normal distribution where 88.3% of the area lies to its left. This is the solution to the problem.

Key Concepts

Normal Distribution

The normal distribution is a continuous probability distribution characterized by its bell-shaped curve, defined by its mean (mu) and standard deviation (sigma). In this distribution, approximately 68% of the data falls within one standard deviation of the mean, 95% within two, and 99.7% within three. Understanding this concept is crucial for interpreting probabilities and areas under the curve.

Z-Score

A Z-score represents the number of standard deviations a data point is from the mean of a distribution. It is calculated using the formula Z = (X - mu) / sigma, where X is the value of interest. Z-scores are essential for finding probabilities associated with specific values in a normal distribution, as they allow for the use of standard normal distribution tables.

Percentiles

A percentile indicates the relative standing of a value within a dataset, representing the percentage of data points that fall below it. For example, if a value is at the 88.3rd percentile, it means that 88.3% of the data lies to the left of that value. Understanding percentiles is key to solving problems that involve finding specific values based on the area under the normal distribution curve.