Question

The time until recharge for a battery in a laptop computer under common conditions is normally distributed

with a mean of 260 minutes and a standard deviation of 50 minutes.
(a) What is the probability that a battery lasts more than four hours?
(b) Calculate the quartiles of battery life. The quartiles are the values q1 and q2 such that the probability
battery lifetime is greater than q1 is 25% (this is called the 75% quartile) and the probability that battery
lifetime is greater than q2 is 75% (this is called the (the 25% quartile)
(c) What value of life in minutes is exceeded with 95% probability?

Answer 1

Explanation:

(a) To calculate the probability that a battery lasts more than four hours, we need to convert four hours into minutes. Since there are 60 minutes in an hour, four hours is equal to 4 * 60 = 240 minutes.

Now, we can standardize the value using the z-score formula:

z = (x - μ) / σ

where x is the value we want to standardize, μ is the mean, and σ is the standard deviation.

z = (240 - 260) / 50 = -0.4

Next, we need to find the probability of the battery lasting more than four hours, which is equivalent to finding the probability of z being greater than -0.4. We can look up this probability in the standard normal distribution table or use a calculator.

Using a standard normal distribution table, we find that the probability corresponding to z = -0.4 is approximately 0.3446.

Therefore, the probability that a battery lasts more than four hours is 0.3446 or 34.46%.

(b) To find the quartiles of battery life, we can use the z-score formula and the cumulative distribution function (CDF) of the normal distribution.

For the 25% quartile (q1), we need to find the value of battery life for which the probability of battery life being greater than q1 is 25%. In other words, we need to find the z-score corresponding to a cumulative probability of 0.25.

Using a standard normal distribution table or a calculator, we find that the z-score corresponding to a cumulative probability of 0.25 is approximately -0.6745.

Now, we can use the z-score formula to find the corresponding value:

-0.6745 = (q1 - 260) / 50

Solving for q1:

q1 - 260 = -0.6745 * 50

q1 - 260 = -33.725

q1 ≈ 260 - 33.725

q1 ≈ 226.275

Therefore, the 25% quartile (q1) of battery life is approximately 226.275 minutes.

For the 75% quartile (q2), we need to find the value of battery life for which the probability of battery life being greater than q2 is 75%. In other words, we need to find the z-score corresponding to a cumulative probability of 0.75.

Using a standard normal distribution table or a calculator, we find that the z-score corresponding to a cumulative probability of 0.75 is approximately 0.6745.

Using the z-score formula:

0.6745 = (q2 - 260) / 50

Solving for q2:

q2 - 260 = 0.6745 * 50

q2 - 260 = 33.725

q2 ≈ 260 + 33.725

q2 ≈ 293.725

Therefore, the 75% quartile (q2) of battery life is approximately 293.725 minutes.

Answer 2

(a) To find the probability that a battery lasts more than four hours (which is 4 hours * 60 minutes = 240 minutes), we need to calculate the area under the normal distribution curve to the right of 240 minutes.

Using z-score formula: z = (X - μ) / σ

where X is the value (240 minutes), μ is the mean (260 minutes), and σ is the standard deviation (50 minutes).

z = (240 - 260) / 50

z = -20 / 50

z = -0.4

Now, we find the probability using the standard normal distribution table or calculator:

P(X > 240 minutes) = P(Z > -0.4)

Using a standard normal distribution table or calculator, we find P(Z > -0.4) ≈ 0.6554.

So, the probability that a battery lasts more than four hours is approximately 0.6554 or 65.54%.

(b) To calculate the quartiles of battery life, we need to find the values q1 and q2.

The z-score corresponding to the 75% quartile (q1) is the value where P(Z > z1) = 0.25.

Using a standard normal distribution table or calculator, we find the z-score for the 75% quartile (q1) ≈ 0.6745.

Now, we find the quartile values:

q1 = μ + z1 * σ

q1 = 260 + 0.6745 * 50

q1 ≈ 295.725

The z-score corresponding to the 25% quartile (q2) is the value where P(Z > z2) = 0.75.

Using a standard normal distribution table or calculator, we find the z-score for the 25% quartile (q2) ≈ -0.6745.

Now, we find the quartile values:

q2 = μ + z2 * σ

q2 = 260 + (-0.6745) * 50

q2 ≈ 224.275

So, the quartiles of battery life are approximately q1 ≈ 295.725 minutes and q2 ≈ 224.275 minutes.

(c) To find the value of battery life in minutes that is exceeded with 95% probability, we need to find the z-score corresponding to the 95th percentile (where P(Z > z) = 0.05).

Using a standard normal distribution table or calculator, we find the z-score for the 95th percentile ≈ 1.645.

Now, we find the value of life in minutes:

X = μ + z * σ

X = 260 + 1.645 * 50

X ≈ 336.25

So, the value of battery life in minutes that is exceeded with 95% probability is approximately 336.25 minutes.

Explanation: