Answer :
To solve the problem of finding the percentage of values in a normal distribution between 16 and 24 using the 68-95-99.7 rule, we need to first understand how this rule applies to a normal distribution.
The 68-95-99.7 rule, also known as the empirical rule, states the following:
[tex]68\%[/tex] of values lie within one standard deviation of the mean.
[tex]95\%[/tex] of values lie within two standard deviations of the mean.
[tex]99.7\%[/tex] of values lie within three standard deviations of the mean.
In this problem, we have:
- A mean ([tex]\mu[/tex]) of [tex]16[/tex].
- A standard deviation ([tex]\sigma[/tex]) of [tex]4[/tex].
To find how much of the data falls between [tex]16[/tex] and [tex]24[/tex], we first determine how many standard deviations [tex]24[/tex] is from the mean.
[tex]\frac{24 - 16}{4} = \frac{8}{4} = 2[/tex]
This means [tex]24[/tex] is two standard deviations above the mean. According to the 68-95-99.7 rule, about [tex]95\%[/tex] of the data is within two standard deviations of the mean.
However, since [tex]16[/tex] is the mean, it is the center of the distribution. This means [tex]24[/tex] is at the point two standard deviations above the mean, and [tex]16[/tex] itself is at the center. Therefore, the area we’re interested in is only on one side of the mean.
To find the percentage of values between [tex]16[/tex] and [tex]24[/tex]:
- Start from the mean and go up to two standard deviations. This accounts for half of the [tex]95\%[/tex] of values that fall between [tex]\mu - 2\sigma[/tex] and [tex]\mu + 2\sigma[/tex].
To calculate this, divide [tex]95\%[/tex] by [tex]2[/tex]:
[tex]\frac{95}{2} = 47.5\%[/tex]
So, the percentage of values between [tex]16[/tex] and [tex]24[/tex] is approximately [tex]47.5\%[/tex].
