Respuesta :
The probability that a randomly selected depth is within one standard deviation of the mean [tex]\boxed{57.6\%}[/tex].
Further Explanation:
Given:
A marine biologist selects water depths from a uniformly distributed collection between [tex]2.00 m[/tex] and [tex]7.00 m[/tex].
Explanation:
The depth follows uniform distribution with upper limit 7 and lower limit 2.
[tex]\boxed{\overline X\sim U\left({a,b}\right)}[/tex]
Here,[tex]\overline X[/tex] is the mean, [tex]a[/tex] is the lower limit and [tex]b[/tex] is the upper limit.
The mean can be calculated as follows,
[tex]\begin{aligned}{\text{Mean}}&=\frac{1}{2}\left({a+b}\right)\\&=\frac{1}{2}\left({2+7}\right)\\&=\frac{9}{2}\\&=4.5\\\end{aligned}[/tex]
The standard deviation of the depth can be calculated as follows,
[tex]\begin{aligned}{\text{Standard deviation}}&=\sqrt{\frac{{{{\left({b-a}\right)}^2}}}{{12}}}\\&=\sqrt{\frac{{{{\left({7-2}\right)}^2}}}{{12}}}\\&=\sqrt{\frac{{{{\left(5\right)}^2}}}{{12}}}\\&=\sqrt{\frac{{25}}{{12}}}\\&=1.44\\\end{aligned}[/tex]
The probability density function can be obtained as follows,
[tex]\begin{aligned}{\text{Pdf}}&=\frac{1}{{b-a}}\\&=\frac{1}{{7-2}}\\&=\frac{1}{5}\\\end{aligned}[/tex]
The probability that a randomly selected depth is within 1 standard deviation of the mean can be expressed as,
[tex]\begin{aligned}{\text{Probability}}&=P\left({\overline X-\sigma<X<\overline X+\sigma}\right)\\&=P\left({4.5-1.44<X<4.5+1.44}\right)\\&=P\left({3.06<X<5.94}\right)\\\end{aligned}[/tex]
Here,[tex]\overline X[/tex] represents the mean and [tex]\sigma[/tex] represents the standard deviation.
The probability that a randomly selected depth is within 1 standard deviation of the mean can be calculated as follows,
[tex]\begin{aligned}{\text{Probability}}&=\int\limits_{3.06}^{5.94}{\frac{1}{5}dx}\\&=\frac{1}{5}\cdot\left.x\right|_{3.06}^{5.94}\\&=\frac{1}{5}\left[{5.94-3.06}\right]\\&=\frac{{2.88}}{5}\\&=0.576\\\end{aligned}[/tex]
The probability that a randomly selected depth is within one standard deviation of the mean [tex]\boxed{57.6\%}[/tex].
Learn more:
1. Learn more about normal distribution https://brainly.com/question/12698949
2. Learn more about standard normal distribution https://brainly.com/question/13006989
3. Learn more about confidence interval of mean https://brainly.com/question/12986589
Answer details:
Grade: College
Subject: Statistics
Chapter: Confidence Interval
Keywords: Z-score, Z-value, marine biologist, experiments, binomial distribution, standard normal distribution, standard deviation, test, measure, probability, low score, mean, repeating, indicated, normal distribution, percentile, percentage, uniformly distributed.
