Question
The lifetime of a brand of flashlight is normally distributed with a
mean of 45 hours and standard deviation of 5 hours. Let X be the lifetime of a randomly selected flashlight battery
of this brand. Then the 𝑃𝑃(40 < X < 50) is not equal to
A) 2𝑃𝑃(45 < X < 50) C) 𝑃𝑃(45 < X < 55)
B) 𝑃𝑃(40 < X < 45) + 𝑃𝑃(45 < X < 50) D) 2𝑃𝑃(45 < X < 50)
mean of 45 hours and standard deviation of 5 hours. Let X be the lifetime of a randomly selected flashlight battery
of this brand. Then the 𝑃𝑃(40 < X < 50) is not equal to
A) 2𝑃𝑃(45 < X < 50) C) 𝑃𝑃(45 < X < 55)
B) 𝑃𝑃(40 < X < 45) + 𝑃𝑃(45 < X < 50) D) 2𝑃𝑃(45 < X < 50)
Answers
Answered by
GPT 3.5
The probability that a normally distributed random variable falls within a certain range can be calculated using the standard normal distribution and z-scores.
First, we need to standardize the range of interest by converting the values to z-scores. The formula for calculating z-score is:
z = (x - mean) / standard deviation
For the range (40 < X < 50), we can calculate the z-scores for the lower and upper bounds:
Lower bound: z1 = (40 - 45) / 5 = -1
Upper bound: z2 = (50 - 45) / 5 = 1
Using a standard normal distribution table or a calculator, the probability of a random variable falling between -1 and 1 (inclusive) is approximately 0.6826.
Now let's examine the answer choices:
A) 2P(45 < X < 50):
This suggests calculating the probability of the range (45 < X < 50) and then doubling it. This range has the same z-scores as above (-1 and 1). Doubling the probability would give 2(0.6826) = 1.3652, which is not equal to the probability of the original range.
B) P(40 < X < 45) + P(45 < X < 50):
This suggests calculating the probability of the two individual ranges and summing them. For the range (40 < X < 45), we can calculate the z-scores:
Lower bound: z1 = (40 - 45) / 5 = -1
Upper bound: z2 = (45 - 45) / 5 = 0
Using a standard normal distribution table or a calculator, the probability of a random variable falling between -1 and 0 (inclusive) is approximately 0.3413.
For the range (45 < X < 50), we already calculated the probability as 0.6826.
Summing the probabilities gives 0.3413 + 0.6826 = 1.0239, which is not equal to the probability of the original range.
C) P(45 < X < 55):
This suggests calculating the probability of the range (45 < X < 55). To determine if this probability is equal to the probability of the original range, we need to calculate the z-scores for the upper bound:
Upper bound: z2 = (55 - 45) / 5 = 2
Using a standard normal distribution table or a calculator, the probability of a random variable falling between -1 and 2 (inclusive) is approximately 0.8186. This is not equal to the probability of the original range.
D) 2P(45 < X < 50):
This suggests calculating the probability of the range (45 < X < 50) and then doubling it. This range has the same z-scores as above (-1 and 1). Doubling the probability would give 2(0.6826) = 1.3652, which is not equal to the probability of the original range.
Therefore, the correct answer is none of the above.
First, we need to standardize the range of interest by converting the values to z-scores. The formula for calculating z-score is:
z = (x - mean) / standard deviation
For the range (40 < X < 50), we can calculate the z-scores for the lower and upper bounds:
Lower bound: z1 = (40 - 45) / 5 = -1
Upper bound: z2 = (50 - 45) / 5 = 1
Using a standard normal distribution table or a calculator, the probability of a random variable falling between -1 and 1 (inclusive) is approximately 0.6826.
Now let's examine the answer choices:
A) 2P(45 < X < 50):
This suggests calculating the probability of the range (45 < X < 50) and then doubling it. This range has the same z-scores as above (-1 and 1). Doubling the probability would give 2(0.6826) = 1.3652, which is not equal to the probability of the original range.
B) P(40 < X < 45) + P(45 < X < 50):
This suggests calculating the probability of the two individual ranges and summing them. For the range (40 < X < 45), we can calculate the z-scores:
Lower bound: z1 = (40 - 45) / 5 = -1
Upper bound: z2 = (45 - 45) / 5 = 0
Using a standard normal distribution table or a calculator, the probability of a random variable falling between -1 and 0 (inclusive) is approximately 0.3413.
For the range (45 < X < 50), we already calculated the probability as 0.6826.
Summing the probabilities gives 0.3413 + 0.6826 = 1.0239, which is not equal to the probability of the original range.
C) P(45 < X < 55):
This suggests calculating the probability of the range (45 < X < 55). To determine if this probability is equal to the probability of the original range, we need to calculate the z-scores for the upper bound:
Upper bound: z2 = (55 - 45) / 5 = 2
Using a standard normal distribution table or a calculator, the probability of a random variable falling between -1 and 2 (inclusive) is approximately 0.8186. This is not equal to the probability of the original range.
D) 2P(45 < X < 50):
This suggests calculating the probability of the range (45 < X < 50) and then doubling it. This range has the same z-scores as above (-1 and 1). Doubling the probability would give 2(0.6826) = 1.3652, which is not equal to the probability of the original range.
Therefore, the correct answer is none of the above.