If a random sample of size n = 65 is drawn, find ¦Ìx, ¦Ò x and P(24 ¡Ü x ¡Ü 26). (Round ¦Ò x to two decimal places and the probability to four decimal places.)
¦Ìx = Correct: Your answer is correct.
To find ¦Ìx (the sample mean), ¦Òx (the standard deviation of the sample mean), and P(24 ¡Ü x ¡Ü 26) (the probability that the sample mean falls between 24 and 26), you need to have the population mean and the population standard deviation. However, these values are not provided in your question.
Assuming that you have the sample data instead, you can calculate the sample mean (¦Ìx) and the sample standard deviation (s) using the given sample data.
Here are the steps to calculate ¦Ìx, ¦Òx, and P(24 ¡Ü x ¡Ü 26):
1. Calculate the sample mean (¦Ìx):
- Add up all the values in the sample (for example, x1 + x2 + x3 + ... + xn).
- Divide the sum by the sample size (n).
- ¦Ìx = (x1 + x2 + x3 + ... + xn) / n
2. Calculate the sample standard deviation (s):
- Subtract the sample mean (¦Ìx) from each value in the sample, square the result, and sum all the squared differences.
- Divide the sum by (n-1).
- Take the square root of the result.
- ¦Òx = √(((x1 - ¦Ìx)^2 + (x2 - ¦Ìx)^2 + (x3 - ¦Ìx)^2 + ... + (xn - ¦Ìx)^2) / (n-1))
3. Calculate the probability P(24 ¡Ü x ¡Ü 26):
- Since the sample mean (¦Ìx) and the population standard deviation are not provided, we can't directly calculate the probability using those values.
- However, if the sample size is large enough (typically n ≥ 30), we can assume that the sample mean follows a normal distribution due to the central limit theorem.
- Assuming the sample mean follows a normal distribution:
- Calculate z-scores for the given values of 24 and 26 using the sample mean (¦Ìx) and the sample standard deviation (s).
- P(24 ¡Ü x ¡Ü 26) can be calculated by finding the area under the normal distribution curve between those two z-scores.
- Use a standard normal distribution table or a calculator to find the corresponding probability.
Remember, without the population mean and the population standard deviation, we can only calculate the sample mean, the sample standard deviation, and make a probabilistic assumption based on the sample size.