The average scuba diver can stay under water for 42 minutes with a standard deviation of 8 minutes.(show graph)
B. what is the probabilty that a diver stays under for the less than 53 minutes?
c.what is the probabilty that a diver stays under between 25 and 50?
d. what is the probabilty that a diver stays under for more than 52 mintues?
e.find the cutoff number for the middle 50% of the time a diver stays down?
f.find the top 15% of time the diver stays down?
g. find the bottom 20%of the time a diver stays down?
To answer these questions, we will use the normal distribution and the Z-score. The Z-score is a measure of how many standard deviations a value is away from the mean.
Step 1: Calculate the Z-score for each question using the formula:
Z = (X - mean) / standard deviation
where X is the given value, mean is the average time (42 minutes), and standard deviation is 8 minutes.
Let's go through each question one by one:
B. To find the probability that a diver stays under 53 minutes, we need to find the area under the curve to the left of 53 minutes.
Step 2: Calculate the Z-score for 53 minutes.
Z = (53 - 42) / 8 = 1.375
Step 3: Refer to a Z-table or use a calculator with the Z-table function to find the corresponding probability. The Z-table will give you the area to the left of the Z-score.
Based on the Z-table, the area to the left of 1.375 is approximately 0.9157.
Therefore, the probability that a diver stays under 53 minutes is 0.9157.
C. To find the probability that a diver stays under between 25 and 50 minutes, we need to find the area under the curve between these two values.
Step 2: Calculate the Z-scores for 25 and 50 minutes.
For 25 minutes:
Z1 = (25 - 42) / 8 = -2.125
For 50 minutes:
Z2 = (50 - 42) / 8 = 1.0
Step 3: Use the Z-table to find the area to the left of Z1 and Z2, and then subtract the smaller area from the larger area to find the probability.
Based on the Z-table, the area to the left of -2.125 is approximately 0.0162, and the area to the left of 1.0 is approximately 0.8413.
Probability = 0.8413 - 0.0162 = 0.8251
Therefore, the probability that a diver stays under between 25 and 50 minutes is 0.8251.
D. To find the probability that a diver stays under for more than 52 minutes, we need to find the area to the right of 52 minutes.
Step 2: Calculate the Z-score for 52 minutes.
Z = (52 - 42) / 8 = 1.25
Step 3: Use the Z-table to find the area to the left of 1.25 and subtract it from 1 to get the area to the right.
Based on the Z-table, the area to the left of 1.25 is approximately 0.8944.
Probability = 1 - 0.8944 = 0.1056
Therefore, the probability that a diver stays under for more than 52 minutes is 0.1056.
E. To find the cutoff number for the middle 50% of the time a diver stays down, we need to find the Z-scores for the upper and lower percentiles.
Step 2: Find the Z-score for the lower percentile.
The lower percentile is (100 - 50) / 2 = 25%. This corresponds to a Z-score of -0.6745.
Step 3: Find the Z-score for the upper percentile.
The upper percentile is (100 + 50) / 2 = 75%. This corresponds to a Z-score of 0.6745.
Step 4: Calculate the cutoff numbers using the Z-scores.
For the lower cutoff number:
X1 = Z1 * standard deviation + mean
X1 = -0.6745 * 8 + 42 = 36.60
For the upper cutoff number:
X2 = Z2 * standard deviation + mean
X2 = 0.6745 * 8 + 42 = 47.40
Therefore, the cutoff number for the middle 50% of the time a diver stays down is between 36.60 and 47.40 minutes.
F. To find the top 15% of time the diver stays down, we need to find the Z-score for the upper 15th percentile.
Step 2: Find the Z-score for the upper percentile.
The upper percentile is 100 - 15 = 85%. This corresponds to a Z-score of 1.0364.
Step 3: Calculate the cutoff number using the Z-score.
X = Z * standard deviation + mean
X = 1.0364 * 8 + 42 = 50.29
Therefore, the top 15% of time the diver stays down is any time above 50.29 minutes.
G. To find the bottom 20% of the time a diver stays down, we need to find the Z-score for the lower 20th percentile.
Step 2: Find the Z-score for the lower percentile.
The lower percentile is 20%. This corresponds to a Z-score of -0.8416.
Step 3: Calculate the cutoff number using the Z-score.
X = Z * standard deviation + mean
X = -0.8416 * 8 + 42 = 34.87
Therefore, the bottom 20% of the time a diver stays down is any time below 34.87 minutes.
I hope this helps! Let me know if you have any further questions.