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.)

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 =

As we know,

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

To find,

The value of Z

So,

μ )/σ

by putting the values using Central Limit Theorem,

∵

Since , the claim is not justified.

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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 and standard deviation , the zscore of a measure X is given by:

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 and standard deviation , the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean and standard deviation .

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:

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.

By the Central Limit Theorem

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.

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step-by-step explanation: