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Answer :
The probability that a single randomly selected value from the population is greater than 102 is approximately 0.4303. The probability that a randomly selected sample of size 21 is approximately 0.0048.
To find the probability that a single randomly selected value is greater than 102, we need to calculate the area under the normal distribution curve to the right of 102. We can use the standard normal distribution table or a calculator to find the corresponding z-score for 102, and then calculate the probability associated with that z-score. The z-score can be calculated using the formula:
z = (x - μ) / σ
where x is the value of interest, μ is the population mean, and σ is the population standard deviation. Plugging in the given values, we have:
z = (102 - 99.9) / 47.6 ≈ 0.0445
Using the z-table or a calculator, we can find that the probability associated with a z-score of 0.0445 is approximately 0.4303. Therefore, the probability that a single randomly selected value is greater than 102 is approximately 0.4303.
To find the probability that a sample of size 21 has a mean greater than 102, we need to consider the sampling distribution of the mean. The mean of the sampling distribution is equal to the population mean, μ, and the standard deviation of the sampling distribution, also known as the standard error, is equal to σ / sqrt(n), where n is the sample size. Plugging in the given values, we have:
standard error = 47.6 / sqrt(21) ≈ 10.3937
Now we can calculate the z-score for a sample mean of 102 using the formula:
= (sample mean - μ) / standard error
Plugging in the values, we have:
z = (102 - 99.9) / 10.3937 ≈ 0.2022
Using the z-table or a calculator, we can find that the probability associated with a z-score of 0.2022 is approximately 0.5793. However, since we are interested in the probability of the sample mean being greater than 102, we need to consider the area under the normal curve to the right of the z-score. Therefore, the probability that a sample of size 21 is randomly selected with a mean greater than 102 is approximately 1 - 0.5793 ≈ 0.4207, or approximately 0.0048 when rounded to four decimals.
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