The lifetime in hours of a certain make of batteries follows a normal distribution with mean 160 and standard deviation 30.

Calculate the interval, symmetrical about the mean within which 75% of the battery lifetimes lie.

The following applet takes the place of your tables or charts traditionally used for this kind of work

http://davidmlane.com/normal.html

click on : value from an area
enter 0.75 in "area"
enter 160 in "mean"
enter 30 in "SD"

under "results" click "between"
click on "recalculate" to get

125.5 to 194.5

Why did the batteries go to therapy?

Because they were feeling a bit "charged" with emotions and needed some positive current to get back on track!

To calculate the interval within which 75% of the battery lifetimes lie, we can use the properties of the normal distribution.

Since the interval is symmetrical about the mean, we need to find the standard deviation that corresponds to the 75th percentile.

To do this, we can use the z-score formula:
z = (X - μ) / σ

where z is the z-score, X is the desired percentile (in this case, 75), μ is the mean, and σ is the standard deviation.

Using the formula, we can solve for the z-score:
75 = (X - 160) / 30

Rearranging the equation, we get:
X - 160 = 30 * 0.674 (the z-score for the 75th percentile)

Simplifying, we find:
X - 160 ≈ 20.22

Adding 160 to both sides, we get:
X ≈ 180.22

Therefore, the interval within which 75% of the battery lifetimes lie is approximately (160 - 20.22, 160 + 20.22), or (139.78, 200.22) hours.

To find the interval within which 75% of the battery lifetimes lie, we need to find the z-score that corresponds to the 75th percentile, and then calculate the corresponding battery lifetime values.

Step 1: Find the z-score
Since the normal distribution is symmetrical, we can find the z-score for the 75th percentile by subtracting 0.75 from 1 (since the area on one side of the mean is 0.5), and then finding the z-score using a standard normal distribution table or calculator.
Z = 1 - 0.75
Z = 0.25

Step 2: Find the x-values
Next, we need to find the corresponding x-values (battery lifetimes) for the z-score we calculated in the previous step. We can use the formula: x = μ + Z * σ, where μ is the mean and σ is the standard deviation.
Using the given values:
x1 = 160 + 0.25 * 30
x1 = 167.5 (rounded to one decimal place)
x2 = 160 - 0.25 * 30
x2 = 152.5 (rounded to one decimal place)

Step 3: Calculate the interval
Finally, we have the interval within which 75% of the battery lifetimes lie.
Interval = [x2, x1]
Interval = [152.5, 167.5] (rounded to one decimal place)

Therefore, the interval, symmetrical about the mean, within which 75% of the battery lifetimes lie is [152.5, 167.5] hours.

To calculate the interval within which 75% of the battery lifetimes lie, we will use the concept of the standard normal distribution.

1. Convert the given values to the standard normal distribution:
Since the lifetime follows a normal distribution with a mean of 160 and a standard deviation of 30, we can calculate the standard score (z-score) for the 75th percentile. The formula for calculating the z-score is:
z = (x - μ) / σ
where x is the percentile value, μ is the mean, and σ is the standard deviation.

For the 75th percentile, x = 75, μ = 160, and σ = 30. Plugging these values into the formula, we get:
z = (75 - 160) / 30
= -85 / 30
= -2.8333 (rounded to four decimal places)

2. Look up the z-score in the standard normal distribution table:
The standard normal distribution table provides the area under the curve to the left of a given z-score. In this case, we are interested in finding the area to the left of -2.8333. By looking up this value in the table, we find that the area is approximately 0.0023.

3. Determine the interval within which 75% of the battery lifetimes lie:
Since the standard normal distribution is symmetrical, we need to find the range from the left to the right tail of the distribution that corresponds to 75% of the area. To do that, we subtract 0.0023 from 0.75 (75%). This gives us:
0.75 - 0.0023 = 0.7477

We need to find the z-score that corresponds to an area of 0.7477 in the standard normal distribution table. By looking for the closest value in the table, we find that it corresponds to a z-score of approximately 0.6745.

4. Convert the z-score back to the original distribution scale:
To convert the z-score back to the original scale, we use the formula:
x = z * σ + μ

Given that z = 0.6745, σ = 30, and μ = 160, we can calculate the interval within which 75% of the battery lifetimes lie:
x = 0.6745 * 30 + 160
= 20.235 + 160
= 180.235

Therefore, 75% of the battery lifetimes lie within the interval (160 - 20.235, 160 + 20.235), which simplifies to (139.765, 180.235) when rounded to three decimal places.

So, the interval, symmetrical about the mean, within which 75% of the battery lifetimes lie is (139.765, 180.235).