A final exam in Math 160 has a mean of 73 with standard deviation 7.73. Assume that a random sample of 24 students is selected and the test score of the sample is computed. Assuming the scores are normally​ distributed, what percentage of sample means are less than 76​? Round the standard score to the nearest tenth before using the​ z-score tables. Round the final answer to the nearest hundredth of a percent.

I get .651 or 65.1 percent

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To find the percentage of sample means that are less than 76, we need to calculate the z-score for 76 and use the z-score table to find the corresponding percentage.

The z-score formula is:
z = (x - μ) / σ

where:
x is the sample mean (76),
μ is the population mean (73), and
σ is the population standard deviation (7.73).

Calculating the z-score:

z = (76 - 73) / 7.73
= 0.38794

Rounding the z-score to the nearest tenth:
z ≈ 0.4

Now, we can use the z-score table to find the percentage of sample means that are less than 76.

Looking up the z-score of 0.4 in the z-score table, we find that the corresponding area to the left of 0.4 is 0.6554.

To convert this to a percentage, we multiply by 100:
0.6554 * 100 ≈ 65.54

Therefore, approximately 65.54% of the sample means are less than 76.

To find the percentage of sample means that are less than 76, we need to convert the test scores to standard scores using the formula:

Z = (X - µ) / σ

Where:
Z is the standard score,
X is the observed score,
µ is the mean of the population,
σ is the standard deviation of the population.

In this case, the mean (µ) is 73 and the standard deviation (σ) is 7.73. We want to find the percentage of sample means less than 76.

First, we need to find the standard score (Z) for 76:
Z = (76 - 73) / 7.73 = 0.39 (rounded to nearest tenth)

Next, we use a standard normal distribution table (also called a z-score table) to find the percentage of values less than 0.39.

Looking up the value 0.39 in the z-score table, we find that the percentage of values less than 0.39 is approximately 0.6517.

However, since we want to find the percentage of sample means (not individual scores), we need to take into account the distribution of sample means. According to the Central Limit Theorem, the distribution of sample means approaches a normal distribution with a mean (µ) equal to the population mean (73 in this case) and a standard deviation (σ) equal to the population standard deviation divided by the square root of the sample size.

The standard deviation of the sample means (σ_m) can be calculated as:
σ_m = σ / sqrt(n)

In this case, the sample size (n) is 24. So we have:
σ_m = 7.73 / sqrt(24) ≈ 1.57

To find the percentage of sample means less than 76, we need to convert this value to a standard score. Using the formula:
Z_m = (X_m - µ) / σ_m

Where X_m is the sample mean (76 in this case), we have:
Z_m = (76 - 73) / 1.57 ≈ 1.91 (rounded to nearest tenth)

Now, we look up the value 1.91 in the z-score table to find the percentage of values less than 1.91. According to the table, this value corresponds to approximately 0.9726.

So, approximately 97.26% of sample means will be less than 76.

Note: These calculations assume that the distribution of test scores is normal and the sample is random.