Let x denote the time it takes to run a road race. suppose x is approximately normally distributed with a mean of 190 minutes and a standard deviation of 21 minutes. If one runner is selected at random, what is the probability that this runner will complete this road race in less than 168 minutes?
To solve this problem, we will convert the given values into z-scores and use the standard normal distribution table.
The z-score formula is given by:
z = (x - μ) / σ
where:
x = value we want to convert to a z-score (168 minutes),
μ = population mean (190 minutes),
σ = population standard deviation (21 minutes).
Substituting the given values:
z = (168 - 190) / 21
z = -22 / 21
z ≈ -1.048
Next, we can use the z-score to find the probability using the standard normal distribution table. However, since we want to find the probability of completing the race in less than 168 minutes, we need to find the area to the left of the z-score.
Using the standard normal distribution table or a calculator, we find that the probability associated with a z-score of -1.048 is approximately 0.1459.
Therefore, the probability that a runner will complete this road race in less than 168 minutes is approximately 0.1459, or 14.59%.
To find the probability that a runner will complete the road race in less than 168 minutes, we need to calculate the cumulative probability up to 168 minutes using the normal distribution.
Step 1: Standardize the value
We will start by standardizing the value 168 using the formula:
z = (x - μ) / σ
Where:
x = 168 minutes (the value we want to find the probability for)
μ = 190 minutes (mean)
σ = 21 minutes (standard deviation)
Substituting the values, we get:
z = (168 - 190) / 21 = -1.048
Step 2: Look up the cumulative probability
Next, we need to find the cumulative probability (area under the curve) to the left of z = -1.048 using a standard normal distribution table or calculator.
Using a standard normal distribution table, the cumulative probability for z = -1.048 is approximately 0.1446.
Step 3: Calculate the probability
The cumulative probability represents the probability that a runner will complete the road race in less than 168 minutes. Hence, the probability is approximately 0.1446 or 14.46%.
Therefore, if one runner is selected at random, the probability that this runner will complete the road race in less than 168 minutes is approximately 0.1446 or 14.46%.
To find the probability that a runner will complete the road race in less than 168 minutes, we need to use the concept of standard normal distribution.
First, we need to standardize the value of 168 minutes using the formula for standardization:
z = (x - μ) / σ
Where:
- x is the value we want to standardize (168 minutes in this case)
- μ is the mean of the distribution (190 minutes)
- σ is the standard deviation of the distribution (21 minutes)
Plugging in the values, we have:
z = (168 - 190) / 21
Calculating this, we get:
z ≈ -1.0476
Next, we need to find the probability associated with this standardized value using a standard normal distribution table or a calculator. The standard normal distribution table provides the area to the left of a given z-value.
Looking up the z-value of -1.0476 in the standard normal distribution table or using a calculator, we find that the corresponding probability is approximately 0.1492.
Therefore, the probability that a runner will complete the road race in less than 168 minutes is approximately 0.1492, or 14.92%.