Question

Given that \(\mathrm{x}\) has a normal distribution with mean 10 and standard deviation 4, find \(\mathrm{P}(\mathrm{x}<15)\).

Short answer

The probability that x is less than 15 for a normal distribution with a mean of 10 and a standard deviation of 4 is approximately 0.8944 or 89.44%.

Step-by-step solution

Identify the given values

Mean (\(\mu\)) = 10, Standard Deviation (\(\sigma\)) = 4, and the value of x for which we want to find the probability, x = 15.

Calculate the z-score

The z-score is a measure of how many standard deviations away x is from the mean. The formula for calculating the z-score is: \[z = \frac {x - \mu} {\sigma}\] Substitute the values in the formula: \[z = \frac {15 - 10} {4}\] Calculate the z-score: \[z = 1.25\]

Find the probability using a z-table or calculator

Now that we have the z-score, we can find the probability using a z-table or calculator. A z-table provides the probability (area under the curve) to the left of a given z-score. In this case, to find \(\mathrm{P}(\mathrm{x}<15)\), we want the probability to the left of the z-score 1.25. Using a z-table or calculator, we can find the probability associated with the z-score 1.25: \(\mathrm{P}(z < 1.25) = 0.8944\)

Interpret the result

The probability that x is less than 15 for a normal distribution with a mean of 10 and a standard deviation of 4 is approximately 0.8944 or 89.44%. This means there is an 89.44% chance that a randomly chosen value from this normal distribution will be less than 15.

Key concepts

Probability Calculation

Calculating probability under a normal distribution can often seem daunting, but it's actually quite straightforward with a step-by-step approach. In a normal distribution, the probability refers to the area under the curve for a given range of values. When you're asked to find a probability, such as \( \mathrm{P}(x < 15) \), you're essentially looking for how likely it is for a randomly selected value to fall below a certain point on a distribution curve.

To figure out this probability, we use a tool known as the z-score. The z-score represents how many standard deviations away a specific value is from the mean. Once you have this number, you can use a z-table or a calculator to find the corresponding probability, providing a percentage indicating the likelihood that a value will be less than the specified point.

In our example, the probability that \(x\) is less than 15 was found to be 0.8944, or 89.44%. This calculation forms the foundation of understanding how likely certain outcomes are within a normal distribution.

Z-Score

The z-score is a concept from statistics that helps us understand where a particular value lies within a normal distribution. It is a standardized score, telling us how many standard deviations a value is from the mean. This concept is key in probability calculations involving normal distributions.

To calculate the z-score, you use the formula:

• \(z = \frac{x - \mu}{\sigma}\)

Where:

• \(x\) is the value in question,
• \(\mu\) is the mean of the distribution,
• \(\sigma\) is the standard deviation.

For instance, with a mean (\(\mu\)) of 10, a standard deviation (\(\sigma\)) of 4, and a value \(x\) of 15, the z-score comes out to be 1.25.

Understanding the z-score allows you to quickly determine where the value sits relative to the distribution's average, and it's your gateway to interpreting probabilities from a z-table.

Standard Deviation

Standard deviation is a measure of spread in a dataset or a probability distribution. It tells you how much the individual data points deviate, or differ, from the mean (average) value. In simpler terms, it gives an idea of the 'spread' of the data.

A small standard deviation means the numbers are closely clustered around the mean, while a large standard deviation indicates a wider spread of numbers.

In the context of the normal distribution exercise, given a standard deviation (\(\sigma\)) of 4, we see that the spread of the distribution is moderately wide. This is crucial for calculating how deviant a value is from the mean, which is where the z-score calculation comes into play. The greater the standard deviation, the less impact any one value has over the mean within the distribution.

Mean

The mean is a central concept in statistics, often referred to as the average. It is the sum of all values divided by the number of values. In normal distributions, the mean is denoted by \(\mu\).

The mean is the point about which the data is balanced; it provides a central, typical value for the dataset or probability distribution. In a symmetric normal distribution, the mean divides the dataset into two equal halves, marking the center point of the bell curve.

In our exercise example, the mean is given as 10. This means most values in the distribution are around 10, assuming it's a perfectly normal distribution. Using this mean in conjunction with the standard deviation allows us to determine the z-score and thereby find probabilities, such as the probability of \(x\) being less than 15.