Male players at the high school, college, and professional ranks use a regulation basketball that weighs 22.0 ounces with a standard deviation of 1.2 ounces. Assume that the weights of basketballs are approximately normally distributed. If a basketball is randomly selected, what is the probability that it will weigh between 21.4 and 23.8 ounces

The probability that a basketball will weigh between 21.4 and 23.8 ounces is 0.62465.

Step-by-step explanation:

We have all this information from the question:

To answer this question:

Important concepts to remember:

For this, it is crucial three concepts: the standard normal distribution, the standard normal table, and z-scores:

Roughly speaking, the standard normal distribution is a normal distribution for standardized values. We can obtain standardized values using the formula for z-scores:

[1]

And these values represent the distance from the population mean in standard deviation units. When they are positive, these values are above the population mean, . In case they are negative, they are below .

We can obtain probabilities for any normally distributed data using the standard normal distribution. These values are tabulated into the standard normal table, available in Statistics books or on the Internet.

In general, these values are cumulative probabilities, that is, probabilities from to the value x in question (a raw value).

At this stage, we have enough information to solve the question.

Solving the question

Cumulative probability for ounces.

That is, the raw score is 0.5 standard deviations below, , the population mean.

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 z. In this case, this second digit is zero (or to be more precise, -0.00), because . With the intersection of these two values in the table, namely, -0.5 and -0.00, we finally obtain the cumulative probability, .

Thus,

Cumulative probability for ounces.

We can follow the same steps as before:

using the standard normal table (z =1.5, +0.00).

Therefore,

Then, to answer the probability that a basketball will weigh between 21.4 and 23.8 ounces, we subtract (as we mentioned before) both cumulative probabilities:

Then, the probability that a basketball will weigh between 21.4 and 23.8 ounces is 0.62465.

We can see this probability represented by the shaded area in the below graph.  

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step-by-step explanation: