The mean time to complete a certain stat quiz is 25 minutes, with a standard deviation of 5 minutes. Find the probability of a student taking.
1.) less than 15 minutes.
2.) Between 15 and 25 minutes.
3.)more than 25 minutes.
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportions/probabilities related to the Z scores.
To find the probabilities for different time intervals, we need to use the standardized normal distribution. In this case, we'll be using the Z-score formula to calculate the probabilities. The Z-score formula is:
Z = (X - μ) / σ
Where:
Z is the Z-score
X is the given time interval
μ is the mean time
σ is the standard deviation
1.) To find the probability of a student taking less than 15 minutes, we need to calculate the Z-score first:
Z = (15 - 25) / 5 = -2
Next, we can consult a Z-table or use a calculator to find the corresponding probability for a Z-score of -2. The Z-table or calculator will give us the value of P(Z < -2). In this case, using a Z-table, we find that P(Z < -2) is approximately 0.0228.
Therefore, the probability of a student taking less than 15 minutes is 0.0228 or 2.28%.
2.) To find the probability of a student taking between 15 and 25 minutes, we need to calculate the Z-scores for both 15 minutes and 25 minutes:
For 15 minutes:
Z₁ = (15 - 25) / 5 = -2
For 25 minutes:
Z₂ = (25 - 25) / 5 = 0
We need to find the probabilities P(Z₁ < Z < Z₂). Again, we can consult a Z-table or use a calculator to find these probabilities.
Using the Z-table, we can find P(Z < -2) which is approximately 0.0228, and P(Z < 0) which is 0.5000. Then, we subtract the smaller probability from the larger probability: P(Z < 0) - P(Z < -2) = 0.5000 - 0.0228 = 0.4772.
Therefore, the probability of a student taking between 15 and 25 minutes is 0.4772 or 47.72%.
3.) To find the probability of a student taking more than 25 minutes, we only need to calculate the Z-score for 25 minutes:
Z = (25 - 25) / 5 = 0
Again, using a Z-table or calculator, we find P(Z > 0) which is 0.5000.
Therefore, the probability of a student taking more than 25 minutes is 0.5000 or 50%.