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Answer :
The final answer is: [tex]\[ \boxed{51.33\%} \][/tex]
To find the percentage of males who would be able to use the boat seat, we need to calculate the proportion of the population that falls within the weight range of 160 to 175 pounds and then convert that proportion to a percentage.
Given that the weight of adult males is normally distributed with a standard deviation of 10 pounds, we first need to find the mean weight. We can calculate the mean from the given data points by summing them up and dividing by the number of data points.
Mean weight [tex]\( \mu \)[/tex] is calculated as follows:
[tex]\[ \mu = \frac{\sum x_i}{n} \][/tex]
where [tex]\( x_i \)[/tex] are the individual weights and [tex]\( n \)[/tex] is the number of weights.
The sum of the given weights is:
[tex]\[ 150 + 152 + 154 + 155 + 156 + 158 + 158 + 160 + 161 + 163 + 164 + 165 + 165 + 166 + 166 + 166 + 167 + 167 + 168 + 169 + 171 + 172 + 174 + 174 + 176 + 177 + 178 + 179 + 179 + 181 + 183 \][/tex]
[tex]\[ = 5070 \][/tex]
The number of weights [tex]\( n \)[/tex] is 31.
So the mean weight [tex]\( \mu \)[/tex] is:
[tex]\[ \mu = \frac{5070}{31} \approx 163.55 \][/tex]
Now, we need to find the [tex]\( z \)[/tex]-scores for the weights 160 and 175 pounds using the formula:
[tex]\[ z = \frac{x - \mu}{\sigma} \][/tex]
where [tex]\( x \)[/tex] is the value for which we want to find the [tex]\( z \)-score, \( \mu \)[/tex] is the mean, and [tex]\( \sigma \)[/tex] is the standard deviation.
For [tex]\( x = 160 \)[/tex]:
[tex]\[ z_{160} = \frac{160 - 163.55}{10} \approx -0.355 \][/tex]
For [tex]\( x = 175 \)[/tex]:
[tex]\[ z_{175} = \frac{175 - 163.55}{10} \approx 1.145 \][/tex]
Next, we find the area under the standard normal distribution curve between these [tex]\( z \)[/tex]-scores using a standard normal distribution table or a calculator.
The area to the left of [tex]\( z_{160} \)[/tex] is approximately 0.3606 and the area to the left of [tex]\( z_{175} \)[/tex] is approximately 0.8739.
The area between the two [tex]\( z \)[/tex]-scores is the difference between these two areas:
[tex]\[ P(160 < x < 175) = P(z < 1.145) - P(z < -0.355) \][/tex]
[tex]\[ \approx 0.8739 - 0.3606 \][/tex]
[tex]\[ \approx 0.5133 \][/tex]
To convert this proportion to a percentage, we multiply by 100:
[tex]\[ 0.5133 \times 100 \approx 51.33\% \][/tex]
Therefore, approximately 51.33% of males would be able to use the boat seat.
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Answer:
80%
Step-by-step explanation:
The number of males with weight between 160 and 175 lbs is 12
So since we were told that the boat seat is graded for males with 160 to 175lbs
=175lbs-160 lbs =15
=12/15*100
=80%
