Given a normal distribution with u=20 and the standard deviation =2.5, how do you find the value of x that has (a) 25% of the distribution's area to the left and (b) 45% of the distributions area to the right?

Answer 1
Firstly we must look at the z-score formula, which is # z= (barx - mu)/sigma#
now we can add what we got into our formula so far. #z = (barx - 20)/(2,5)#
now in question a it tells us that #Phi(z)= 0.25# (note that they said #25%# to the left)
#Phi# is the symbol to say you using the CDF of the normal distribution
So to solve for #z# we can just use the z-score table, which we will get. (The table gives us area, or probability to the left)
Thus #z ~~ -0.675#

then we plot into our formula.

# -0.675 = (barx - 20)/2.5#
then we solve to get #barx#
# barx = 18.3125#

now to go to question b

note that in section b here they ask for #45%# to the right of the point, and our tables gives us to the left of the point.
as our table is using a CDF we know the total area underneat the curve is going to equal #1# which will leave us with the sum.
#1 - Phi(z) = 0.45# so we actually look for #Phi(z) = 0.55#
Thus we get that #z ~~ 0.13# by using our table.

put the value into our formula and we get.

#0.13 = (barx-20)/2.5#
then we solve for #barx#
#barx = 20.325#
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Answer 2

(a) To find the value of ( x ) with 25% of the normal distribution's area to the left, we use the z-score formula. The z-score ( z ) is calculated as:

[ z = \frac{x - \mu}{\sigma} ]

where ( \mu ) is the mean (20 in this case) and ( \sigma ) is the standard deviation (2.5 in this case). Since we want 25% of the area to the left, the z-score corresponds to the 25th percentile, denoted as ( z_{0.25} ) in standard normal distribution tables.

Look up the value of ( z_{0.25} ) in the standard normal distribution table. For example, if you have access to statistical software or a z-table, you might find that ( z_{0.25} ) is approximately -0.6745.

Now, substitute the known values into the z-score formula to find ( x ):

[ -0.6745 = \frac{x - 20}{2.5} ]

Solve for ( x ):

[ x = -0.6745 \times 2.5 + 20 ]

Calculate the value of ( x ).

(b) Similarly, to find the value of ( x ) with 45% of the distribution's area to the right, we use the z-score formula and look for the z-score corresponding to the 55th percentile, denoted as ( z_{0.55} ).

Find ( z_{0.55} ) in the standard normal distribution table. Let's assume ( z_{0.55} ) is approximately 0.1257 (this value will depend on the table or software used).

Substitute the known values into the z-score formula:

[ 0.1257 = \frac{x - 20}{2.5} ]

Solve for ( x ) to find the value with 45% of the area to the right.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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