Math  /  Data & Statistics

QuestionThe scores of the students on a standardized test are normally distributed, with a mean of 500 and a standard deviation of 110 . What is the probability that a randomly selected student has a score between 350 and 550 ? Use the portion of the standard normal table below to help answer the question. \begin{tabular}{|c|c|} \hlinezz & Probability \\ \hline 0.00 & 0.5000 \\ \hline 0.25 & 0.5987 \\ \hline 0.35 & 0.6368 \\ \hline 0.45 & 0.6736 \\ \hline 1.00 & 0.8413 \\ \hline 1.26 & 0.8961 \\ \hline 1.35 & 0.9115 \\ \hline 1.36 & 0.9131 \\ \hline \hline \end{tabular} 9%9 \% 24%24 \% 59%59 \%

Studdy Solution

STEP 1

What is this asking? What's the chance a student's test score lands between 350 and 550, if the average score is 500 and scores typically spread out by 110? Watch out! Don't forget to convert the raw scores to *z*-scores before using the *z*-table!

STEP 2

1. Calculate the *z*-scores.
2. Find the probabilities from the *z*-table.
3. Calculate the probability between the *z*-scores.

STEP 3

Let's **transform** our raw scores into *z*-scores!
This tells us how far each score is from the mean, in terms of standard deviations.
The formula for the *z*-score is z=xμσz = \frac{x - \mu}{\sigma}, where xx is the raw score, μ\mu is the mean, and σ\sigma is the standard deviation.

STEP 4

For x=350x = 350, we have z=350500110=1501101.36z = \frac{350 - 500}{110} = \frac{-150}{110} \approx -1.36.
So, a score of 350 is about **1.36 standard deviations below** the mean.

STEP 5

For x=550x = 550, we have z=550500110=501100.45z = \frac{550 - 500}{110} = \frac{50}{110} \approx 0.45.
This means a score of 550 is about **0.45 standard deviations above** the mean.

STEP 6

Our *z*-table gives us the probability of a score being *less than* a given *z*-score.
For z=0.45z = 0.45, the table tells us the probability is **0.6736**.

STEP 7

For z=1.36z = -1.36, we need to be a little clever.
The table doesn't directly give us probabilities for negative *z*-scores.
However, the normal distribution is symmetric, so the area to the left of z=1.36z = -1.36 is the same as the area to the *right* of z=1.36z = 1.36.

STEP 8

The table tells us the area to the *left* of z=1.36z = 1.36 is **0.9131**.
Since the total area under the curve is 1, the area to the *right* of z=1.36z = 1.36 (and therefore to the left of z=1.36z = -1.36) is 10.9131=0.08691 - 0.9131 = 0.0869.

STEP 9

We want the probability of a score being between z=1.36z = -1.36 and z=0.45z = 0.45.
This is the area under the curve between these two *z*-scores.

STEP 10

We know the area to the left of z=0.45z = 0.45 is **0.6736**, and the area to the left of z=1.36z = -1.36 is **0.0869**.
The area between them is the difference: 0.67360.0869=0.58670.6736 - 0.0869 = 0.5867.

STEP 11

The probability that a randomly selected student has a score between 350 and 550 is approximately 0.59\textbf{0.59} or \textbf{59%}.

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