Q1) The life of an Appex brand battery is 500 days with a standard deviation of 12 days. What percent of values are between 488 and 524?
Q2)The life of an Appex brand battery is 500 days with a standard deviation of 12 days. What percent of values are less than 512?
Q3)The life of an Appex brand battery is 500 days with a standard deviation of 12 days. What percent of values are less than 488?
Q4) The life of an Appex brand battery is 500 days with a standard deviation of 12 days. Calculate the z-score for a battery that lasted 465 days.
You can play around with Z table stuff at
davidmlane.com/hyperstat/z_table.html
To answer these questions, we will use the concept of the normal distribution and z-scores. Here's how you can find the answers:
1) To find the percentage of values between 488 and 524, we need to find the z-scores corresponding to these values and then use a standard normal distribution table (also known as the z-table).
First, we calculate the z-score for the lower value of 488:
z1 = (488 - 500) / 12 = -1.00
Then, we calculate the z-score for the upper value of 524:
z2 = (524 - 500) / 12 = 2.00
Next, we use the z-table to find the area under the standard normal distribution curve between these two z-scores. We subtract the cumulative probability corresponding to z1 from the cumulative probability corresponding to z2.
Using the z-table, we find:
P(z < -1.00) = 0.1587
P(z < 2.00) = 0.9772
The percentage of values between 488 and 524 is:
P(-1.00 < z < 2.00) = P(z < 2.00) - P(z < -1.00) = 0.9772 - 0.1587 = 0.8185 or 81.85%
2) To find the percentage of values less than 512, we need to find the z-score corresponding to 512 and then use the z-table.
The z-score for 512 is:
z = (512 - 500) / 12 = 1.00
Using the z-table, we find:
P(z < 1.00) = 0.8413
The percentage of values less than 512 is 84.13%.
3) To find the percentage of values less than 488, we again find the corresponding z-score and use the z-table.
The z-score for 488 is:
z = (488 - 500) / 12 = -1.00
Using the z-table, we find:
P(z < -1.00) = 0.1587
The percentage of values less than 488 is 15.87%.
4) To calculate the z-score for a battery that lasted 465 days, we use the formula:
z = (x - μ) / σ
where x is the value (465), μ is the mean (500), and σ is the standard deviation (12).
The z-score is:
z = (465 - 500) / 12 = -2.92