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Answer :
The probability that the average height is between 174cm and 176cm is 0.
The average height of freshmen is given as 175 cm, with a standard deviation of 6 cm. We are asked to find the probability that the average height of a random sample of 100 freshmen falls between 174 cm and 176 cm.
To solve this problem, we need to use the concept of the sampling distribution of the sample mean. The sampling distribution of the sample mean follows a normal distribution with a mean equal to the population mean and a standard deviation equal to the population standard deviation divided by the square root of the sample size.
In this case, the population mean is 175 cm and the population standard deviation is 6 cm. Since the sample size is 100, the standard deviation of the sampling distribution of the sample mean would be 6 cm divided by the square root of 100, which is 0.6 cm.
Now, we can convert the given values of 174 cm and 176 cm into z-scores using the formula:
z = (x - μ) / σ
where x is the value, μ is the mean, and σ is the standard deviation.
For 174 cm:
z1 = (174 - 175) / 0.6 = -1.67
For 176 cm:
z2 = (176 - 175) / 0.6 = 1.67
Next, we need to find the probability of the average height falling between these z-scores. We can use the standard normal distribution table or a calculator to find these probabilities.
From the standard normal distribution table, the probability corresponding to a z-score of -1.67 is 0.0475, and the probability corresponding to a z-score of 1.67 is also 0.0475.
Therefore, the probability that the average height of a random sample of 100 freshmen falls between 174 cm and 176 cm is the difference between these two probabilities:
0.0475 - 0.0475 = 0
So, the probability is 0, indicating that the chance of the average height falling between 174 cm and 176 cm is extremely low.
Learn more about probability and standard distribution at:
https://brainly.com/question/27701525.
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