# 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?

then we plot into our formula.

now to go to question b

put the value into our formula and we get.

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(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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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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