Answer:
The probability that a basketball will weigh between 21.4 and 23.8 <em>ounces</em> is 0.62465.
Step-by-step explanation:
<em>We have all this information from the question</em>:
- The weights of the basketballs are <em>approximately normally distributed</em>.
- The population mean,
, for basketball weights is 22.0 ounces,
ounces. - The population standard deviation,
, for basketball weights is 1.2 ounces,
ounces.
<em>To answer this question</em>:
- First, we need to calculate the cumulative probability for
ounces and
ounces. - Second, subtract both values to obtain the asked probability, that is the probability that [a basketball] will weigh between 21.4 and 23.8 ounces.
<em>Important concepts to remember</em>:
For this, it is crucial three concepts: the <em>standard normal distribution, the standard normal table,</em> and <em>z-scores</em>:
Roughly speaking, the <em>standard normal distribution</em> is a normal distribution for <em>standardized values</em>. We can obtain standardized values using the formula for <em>z-scores</em>:
[1]
And these values represent the distance from the population mean in standard deviation units. When they are <em>positive</em>, these values are <em>above</em> the population mean,
. In case they are <em>negative</em>, they are <em>below</em>
.
We can obtain <em>probabilities</em> for any <em>normally distributed data</em> using the <em>standard normal distribution</em>. These values are tabulated into the <em>standard normal table</em>, available in Statistics books or on the Internet.
In general, these values are <em>cumulative probabilities</em>, that is, probabilities from
to the value <em>x</em> in question (a raw value).
At this stage, we have enough information to solve the question.
Solving the question
<em>Cumulative probability for </em>
<em> ounces</em>.
- Obtain the z-score, using [1], for
ounces (without using units):



That is, the raw score
is <em>0.5 standard deviations below, </em>
<em>, </em>the population mean.
- Getting
using the standard normal table.
Since
, we can consult the standard normal table, using
as an entry (using its first column).
The first row of this table has a second digit in the decimal part for the value of <em>z</em>. In this case, this second digit is zero (or to be more precise, -0.00), because
. With the <em>intersection</em> of these <em>two values</em> in the table, namely, -0.5 and -0.00, we finally obtain the cumulative probability,
.
Thus, 
<em>Cumulative probability for </em>
<em> ounces</em>.
We can follow the <em>same steps</em> as before:




using the standard normal table (z =1.5, +0.00).
Therefore, 
Then, to answer the probability that a basketball will weigh between <em>21.4</em> and <em>23.8</em> ounces, we subtract (as we mentioned before) both cumulative probabilities:

Then, the probability that a basketball will weigh between 21.4 and 23.8 <em>ounces</em> is 0.62465.
We can see this probability represented by the shaded area in the below graph.