Question

The average math SAT score is 511 with a standard deviation of 119. A particular high school claims that its students have unusually high math SAT scores. A random sample of 55 students from this school was​ selected, and the mean math SAT score was 528. Is the high school justified in its​ claim? Explain. ▼ No Yes ​, because the​ z-score ​( nothing​) is ▼ unusual not unusual since it ▼ does not lie lies within the range of a usual​ event, namely within ▼ 1 standard deviation 2 standard deviations 3 standard deviations of the mean of the sample means. ​(Round to two decimal places as​ needed.)

Answer 1

Answer:

No, because the​ z-score of Z = 1.06 is not unusual since it does not lie within the range of a usual​ event, namely within 2 standard deviations of the mean of the sample means.

Step-by-step explanation:

To solve this question, we need to understand the normal probability distribution and the central limit theorem.

Normal probability distribution

When the distribution is normal, we use the z-score formula.

In a set with mean $\mu$ and standard deviation $\sigma$, the zscore of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

Central Limit Theorem

The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean $\mu$ and standard deviation $\sigma$, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean $\mu$ and standard deviation $s = \frac{\sigma}{\sqrt{n}}$.

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

Unusual

If X is more than two standard deviations from the mean, x is considered unusual.

In this question:

$\mu = 511, \sigma = 119, n = 55, s = \frac{119}{\sqrt{55}} = 16.046$

A random sample of 55 students from this school was​ selected, and the mean math SAT score was 528. Is the high school justified in its​ claim?

If Z is equal or greater than 2, the claim is justified.

Lets find Z.

$Z = \frac{X - \mu}{\sigma}$

By the Central Limit Theorem

$Z = \frac{X - \mu}{s}$

$Z = \frac{528 - 511}{16.046}$

$Z = 1.06$

1.06 < 2, so 528 is not unusually high.

The answer is:

No, because the​ z-score of Z = 1.06 is not unusual since it does not lie within the range of a usual​ event, namely within 2 standard deviations of the mean of the sample means.

Answer 2

The statement that could be made regarding the high school about the justification of its claim would be:

- No, because the​ z-score of Z = 1.06 is not unusual since it does not lie within the range of a usual​ event, namely within 2 standard deviations of the mean of the sample means.

Given that,

μ = 511

σ = 119

Sample(n) = 55

and

s = $119/\sqrt{55}$

$= 16.046$

As we know,

The claim of the high school could be valid and justified only when

$Z > 2$

To find,

The value of Z

So,

$Z = (X -$ μ )/σ

by putting the values using Central Limit Theorem,

$Z = (528 - 511)/16.046$

∵ $Z = 1.06$

Since $Z < 2$, the claim is not justified.

Learn more about "Standard Deviation" here:

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