John drives to work each morning and the trip takes an average of = 38 minutes. The distribution of driving times is approximately normal with a standard deviation of = 5 minutes. For a randomly selected morning, what is the probability that John's drive to work will take between 36 and 40 minutes?
Use z-scores.
Find two z-scores using this formula:
z = (x - mean)/sd
Mean = 38
Sd = 5
Use 36 for x to determine first z-score.
Use 40 for x to determine second z-score.
Once you have both z-scores, check a z-table for the probability between the two scores.
I hope this will help get you started.
To find the probability that John's drive to work will take between 36 and 40 minutes, we need to first calculate the z-scores for these two values. The z-score measures the number of standard deviations a particular value is away from the mean.
The formula for calculating the z-score is:
z = (x - μ) / σ
Where:
x = The value we want to find the probability for (in this case, 36 and 40)
μ = The mean (average) of the distribution (in this case, 38)
σ = The standard deviation of the distribution (in this case, 5)
For 36 minutes:
z = (36 - 38) / 5 = -0.4
For 40 minutes:
z = (40 - 38) / 5 = 0.4
Now that we have calculated the z-scores, we can use a standard normal distribution table or a calculator to find the corresponding probabilities.
Using a standard normal distribution table, you can look up the probabilities for the z-scores -0.4 and 0.4 separately. The table will give you the area under the standard normal curve for those z-scores.
Alternatively, you can use a calculator or statistical software to find the probabilities directly. For example, using a calculator, you can use the "normalcdf" function to find the probability between the two z-scores -0.4 and 0.4.
P(-0.4 < z < 0.4) ≈ 0.3446
Therefore, the probability that John's drive to work will take between 36 and 40 minutes is approximately 0.3446, or 34.46%.
To find the probability that John's drive to work will take between 36 and 40 minutes, we need to calculate the z-scores for these two values and then find the corresponding probabilities from the standard normal distribution.
Step 1: Calculate the z-score for 36 minutes.
The formula for calculating the z-score is:
z = (x - μ) / σ
where:
x = observed value
μ = mean
σ = standard deviation
Using the provided information:
x = 36 minutes
μ = 38 minutes
σ = 5 minutes
Plugging these values into the formula:
z1 = (36 - 38) / 5 = -0.4
Step 2: Calculate the z-score for 40 minutes.
Using the same formula:
x = 40 minutes
Plugging in the values:
z2 = (40 - 38) / 5 = 0.4
Step 3: Find the probabilities corresponding to these z-scores.
To find the probabilities corresponding to these z-scores, we can refer to a standard normal distribution table or use a calculator.
From the table or calculator, we find that the probability corresponding to a z-score of -0.4 is approximately 0.3446, and the probability corresponding to a z-score of 0.4 is also approximately 0.3446.
Step 4: Calculate the probability between 36 and 40 minutes.
Since we are looking for the probability between 36 and 40 minutes, we need to find the difference between the probabilities corresponding to the z-scores of 36 and 40 minutes.
P(36 ≤ x ≤ 40) = P(z1 ≤ z ≤ z2)
= P(z ≤ 0.4) - P(z ≤ -0.4)
= 0.3446 - 0.3446
= 0
Therefore, the probability that John's drive to work will take between 36 and 40 minutes is approximately 0 (or 0%).