The time until recharge for a battery in a laptop computer under common conditions is normally distributed with mean of 270 minutes and a standard deviation of 50 minutes.
a) What is the probability that a battery lasts more than four hours? Entry field with incorrect answer (Round the answer to 3 decimal places.)
b) What are the quartiles (the 25% and 75% values) of battery life?
25% value (Round the answer to the nearest integer.)
75% value (Round the answer to the nearest integer.)
c) What value of life in minutes is exceeded with 95% probability? Entry field with incorrect answer (Round the answer to the nearest integer.)
http://davidmlane.com/hyperstat/z_table.html
a) To find the probability that a battery lasts more than four hours, we need to convert four hours to minutes. There are 60 minutes in an hour, so four hours is equal to 4 * 60 = 240 minutes.
Given that the time until recharge follows a normal distribution with a mean of 270 minutes and a standard deviation of 50 minutes, we can use the z-score formula to find the probability.
The z-score formula is given by z = (x - mean) / standard deviation.
So, for a battery to last more than four hours (240 minutes), we calculate the z-score as follows:
z = (240 - 270) / 50 = -0.6
Now, we need to find the probability of a z-score being greater than -0.6. We can look this up in the standard normal distribution table or use statistical software. This probability is approximately 0.7257.
Therefore, the probability that a battery lasts more than four hours is 0.725.
b) The quartiles represent the values below which a certain percentage of the data falls. To find the quartiles of battery life, we can use the properties of the normal distribution.
The 25% quartile corresponds to the z-score that leaves 25% of the data to the left. The 75% quartile corresponds to the z-score that leaves 75% of the data to the left.
Using a standard normal distribution table or statistical software, we can find the z-scores associated with the 25% and 75% quartiles. The z-score associated with the 25% quartile is approximately -0.674 and the z-score associated with the 75% quartile is approximately 0.674.
Now, we can calculate the corresponding battery life values using the z-score formula:
For the 25% quartile:
25% value = mean + (z * standard deviation) = 270 + (-0.674 * 50) = 236.3
For the 75% quartile:
75% value = mean + (z * standard deviation) = 270 + (0.674 * 50) = 303.7
Therefore, the 25% value (rounded to the nearest integer) is 236, and the 75% value (rounded to the nearest integer) is 304.
c) To find the value of battery life in minutes that is exceeded with 95% probability, we need to calculate the z-score that leaves 5% of the data to the left.
Using a standard normal distribution table or statistical software, we find that the z-score associated with the 5% probability is approximately -1.645.
Using the z-score formula, we can calculate the corresponding battery life value:
95% value = mean + (z * standard deviation) = 270 + (-1.645 * 50) = 190.775
Therefore, the value of battery life in minutes that is exceeded with 95% probability (rounded to the nearest integer) is 191.
To solve these questions, we can use the concepts of the normal distribution and z-scores. A z-score measures the number of standard deviations a data point is from the mean.
a) To find the probability that a battery lasts more than four hours (which is 240 minutes), we need to convert 240 minutes into a z-score and then find the corresponding area in the tail of the distribution.
First, we calculate the z-score using the formula: z = (x - mean) / standard deviation. In this case, x = 240 minutes, mean = 270 minutes, and standard deviation = 50 minutes.
z = (240 - 270) / 50 = -30 / 50 = -0.6
Next, we find the area to the right of this z-score in a standard normal distribution table or using technology. The area represents the probability.
Using a standard normal distribution table or calculator, we find that the area to the right of -0.6 is approximately 0.7257.
However, since we're interested in the probability that the battery lasts longer than four hours, which is the tail area to the right, we need to subtract this value from 1.
Probability = 1 - 0.7257 = 0.2743
Therefore, the probability that a battery lasts more than four hours is 0.274.
b) To find the quartiles (the 25% and 75% values) of battery life, we need to find the corresponding z-scores. The z-score for the 25th percentile is -0.674 and the z-score for the 75th percentile is 0.674.
To find the actual values, we use the formula: x = mean + (z * standard deviation).
For the 25th percentile, x = 270 + (-0.674 * 50) ≈ 238.63.
For the 75th percentile, x = 270 + (0.674 * 50) ≈ 301.39.
Rounding to the nearest integer, the 25% value would be 239 and the 75% value would be 301.
c) To find the value of battery life in minutes that is exceeded with 95% probability, we need to find the corresponding z-score for the 95th percentile, which is approximately 1.645.
Using the formula: x = mean + (z * standard deviation).
x = 270 + (1.645 * 50) ≈ 349.25.
Rounding to the nearest integer, the value of battery life that is exceeded with 95% probability is 349 minutes.