Statistics HW Help?

a.

What is the probability that all five anemones have a basal diameter more than 5.5

cm?

(2pt)

Here you find the probability of one anemone is greater than 5.5

then you use binomial probability to

find the probability of all 5 being greater.

for one anemones

P( X> 5.5) = 1- P(x< 5.5)

P(X< 5.5 ) = P ( Z < ( x - mean) /standard deviation ) = P (z< (5.5 -5.3) / 1.8) =

P(X< 5.5) = P( z < 0.2/1.8) = P(z< (1/9) ) = P(z< 0.111111111 )

https://www.stat.tamu.edu/~lzhou/stat302/standardn...

Per this table,

P(z< 0.11) = .54380

P(z< 0.12 ) = .54776

=== Interpolating (Some professor and classes tell just the closest value. so if you not supposed

to interpolating use 0.54380 )

=+0.54380 + ( 0.54776-0.54380)*(1/9) = 0.54424

P(x< 5.8) = 0.54424

P(x> 5.8) = 1 - 0.54424 = 0.45576

Now remember binomial probability

P( k successes in n trials ) = (n k ) *(p)^k*(1-p)^(n-k)

where (n k ) = (n!) / ( (n-k)! * k!) =

but we are looking at (n = 5 , k = 5, p = 0.45576)

(5 5) = 5! / ( 5!) *(0!) = 1

===== answer part (a)

P( 5 successes ) = p^k = (0.45576)^5 = 0.019664415

b.

What is the probability that the mean basal diameter of the five anemones is more

than 5.5 cm?

Since the original distribution is normal and you know the population standard deviation .

You can use the normal distribution for the sample mean here.

mean of the sample mean = population mean = 5.3

standard deviation of the sample mean = population standard deviation / sqrt (N)

standard deviation of the sample mean = 1.8/sqrt(5) = 0.804984472

so now use standard equation with mean = 5.3 and standard deviation = 0.804984472

P(x> 5.5) = 1 - P(x<5.5)

P(x< 5.50) = p(Z < (5.5 -5.3) / 0.804984472 ) = p(Z< 0.248451997)

P(z< 0.24 ) = .59483

P(z< 0.25 ) = .59871

so if you interpolate

=0.59483 + ( 0.248451997 - 0.24 ) * ( 0.59871 -0.59483)/ 0.01 =

= 0.598109375

rounding the 5 digits of the table

= 0.59811

P(x> 5.8) = 1-0.59811 = 0.401890625

==== answer (B)

=0.40189

Extra Question

(b) is larger obviously

why is b always larger

for (a) , you are taking your answer for one trial and then raising it to 5th power .

Whenever you square any number which is between 0 and 1 , the result is a smaller number.

Probability always only go from 0 to 1 ( 0 and 1 included). However , the 0 and 1 cases are

really abnormal conditions.

This is even worse when you take a number between 0 and 1 and take it to the 5th power.

Each times you multiple the probability against itself, the results get smaller.

So the answer for (a) will always be smaller.

If the question was P(x< 5.5) , it would be even simpler, but since it is P(x> 5.5) it makes it a little

more complicated.