Question 1207035
you can use the poisson distribution calculator at <a href = "" target = "_blank"></a> to solve parts (a) through (c).
an alternative is to use the poisson distribution table.


here's a reference on the poisson distribution itself.
<a href = "https://www.youtube.com/watch?v=zA7fp2s7FlM" target = "_blank">https://www.youtube.com/watch?v=zA7fp2s7FlM</a>


here's a link to the calculator.
<a href = "https://stattrek.com/online-calculator/poisson#google_vignette" target = "_blank">https://stattrek.com/online-calculator/poisson#google_vignette</a>


here are the results from using the calculator.


<img src = "http://theo.x10hosting.com/2024/043001.jpg">


you have to set the time interval to be consistent.
you were given 48 births in a 24 hour period.
since you want number per hour, you have to set the average to per hoursly.
48 per 24 hours is equivalent to 2 per hour because the hospital is open 24 hours a day.
so the average  is 2 per hour and the variable you are looking for is 3.
you enter 2 for the average and 3 for the number you are looking for and the calculator gives you a bunch of statistics.
since you are looking for x < 3, you choose that.
you get p(x<3) = 0.67668.


part (a) answer is 48.
part (b) answer is 2.
part (c) answer is .67668.


you can also use the poisson dictribution table for cumulative results.
one such table can be found at <a href = "https://ghaidab.weebly.com/uploads/1/1/2/9/11294132/poisson_cdf_table.pdf" target = "_blank">https://ghaidab.weebly.com/uploads/1/1/2/9/11294132/poisson_cdf_table.pdf</a>


lambda is the mean of the distribution, so for this problem, you would look for a lambda of 2 and a value of x as 2.
you are looking for p(x) < 3 which is the same as p(x) <= 2.
p(x) <= is what the table is set up to look for.
you should find that p(x <= 2) = .6767 from the table.
this agrees with what the calculator provided as p(x) < 3.


if you wanted to find p(x) > 3, you would look for 1 - p(x) <= 3.
that would be equal to 1 minus p(x <= 3) = 1 - .0.85712 = .14288.
if you're using the calculator, it tells you othat straight away.
if you're using the table, you look for p(x) <= 3 and take 1 minus that.


the calculator goes to 5 decimal places.
the table goes to 4.


for part (d), you need to use chebyshev's theorem.
that's the way the internet spells it, i believe.
that theorem states that the smallest probability of any type of distribution is given by the equation of p(|x-m|>=k*s) <= 1 - 1/k^2.
this says that the probability that the variable value minus the mean is greater than or equal to k * the standard deviation is smaller than or equal to 1 - 1/k^2.


k is the number of standard deviations above the mean.
the mean is assumed to be in the middle of the range, so the standard deviation is the same whether above or below the mean.
the absolute value symbol takes care of that by making (x-s) always positive.


in your problem, mean is 2 and variance is also 2, since this is a property of the poisson distribution.
standard deviation is therefore sqrt(2).



at least 89% of the time means the probability that it is greater than or equal to 89% = .89
.89 is the minimum proportion.
the formula is p(|x-m| >= k*s) >= 1 - 1/k^2 = .89
add .89 to both sides of the equation abd add 1/k^2 to both sides of the equation to get 1.89 = 1/k^2.
solve for k^2 to get k^2 = 1/1.89.
solve for k to get k = sqrt(1/1.89) = .72739
the interpretatio9n of that would be that:
the minimum percentage of values that are .72739 * the standard deviation about the mean is equal to .89.


here's a reference on chebyshev's formula.
<a href = "https://statisticsbyjim.com/basics/chebyshevs-theorem-in-statistics/" target = "_blank">https://statisticsbyjim.com/basics/chebyshevs-theorem-in-statistics/</a>


the reference states that the maximum proportion = 1/k^2 and the minimum proportion is 1 - 1/k^2.


we used the minimum proportion to solve your problem.