Answer :
First, we calculate the [tex]$z$[/tex]-score for a weight of [tex]$170$[/tex] pounds using the formula
[tex]$$
z = \frac{x - \mu}{\sigma},
$$[/tex]
where [tex]$\mu = 165$[/tex] pounds, [tex]$\sigma = 9.8$[/tex] pounds, and [tex]$x = 170$[/tex] pounds. Thus,
[tex]$$
z = \frac{170 - 165}{9.8} \approx 0.5102.
$$[/tex]
Next, using a [tex]$z$[/tex]-table or appropriate technology, we find the cumulative probability corresponding to [tex]$z = 0.5102$[/tex]. This probability is approximately [tex]$0.6950$[/tex], which means about [tex]$69.5\%$[/tex] of the members are expected to weigh [tex]$170$[/tex] pounds or less.
Since the league has [tex]$150$[/tex] members, we multiply the cumulative probability by the total number of members:
[tex]$$
\text{Number of members} \approx 0.6950 \times 150 \approx 104.26.
$$[/tex]
Rounding to the nearest whole number, about [tex]$104$[/tex] members weigh [tex]$170$[/tex] pounds or less.
[tex]$$
z = \frac{x - \mu}{\sigma},
$$[/tex]
where [tex]$\mu = 165$[/tex] pounds, [tex]$\sigma = 9.8$[/tex] pounds, and [tex]$x = 170$[/tex] pounds. Thus,
[tex]$$
z = \frac{170 - 165}{9.8} \approx 0.5102.
$$[/tex]
Next, using a [tex]$z$[/tex]-table or appropriate technology, we find the cumulative probability corresponding to [tex]$z = 0.5102$[/tex]. This probability is approximately [tex]$0.6950$[/tex], which means about [tex]$69.5\%$[/tex] of the members are expected to weigh [tex]$170$[/tex] pounds or less.
Since the league has [tex]$150$[/tex] members, we multiply the cumulative probability by the total number of members:
[tex]$$
\text{Number of members} \approx 0.6950 \times 150 \approx 104.26.
$$[/tex]
Rounding to the nearest whole number, about [tex]$104$[/tex] members weigh [tex]$170$[/tex] pounds or less.
