To find the probability that the sample mean is between 45 and 55, we can use the Central Limit Theorem (CLT). According to the CLT, the distribution of the sample means approaches a normal distribution as the sample size increases, regardless of the shape of the population distribution. Given that the population mean ($\mu$) is 50 and the population standard deviation ($\sigma$) is 3.6, we can proceed as follows:

Steps:

1. Population mean ($\mu$) = 50

2. Population standard deviation ($\sigma$) = 3.6

3. Sample size (n) = 10

4. The mean of the sampling distribution of the sample mean is equal to the population mean, so: $\mu_{\bar{x}} = \mu = 50$

5. The standard deviation of the sampling distribution (also called the standard error of the mean) is given by: $\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} = \frac{3.6}{\sqrt{10}} \approx 1.139$

6. Now, we standardize the values 45 and 55 using the z-score formula: $z = \frac{x - \mu_{\bar{x}}}{\sigma_{\bar{x}}}$

   For $x = 45$: $z_{45} = \frac{45 - 50}{1.139} = \frac{-5}{1.139} \approx -4.39$

   For $x = 55$: $z_{55} = \frac{55 - 50}{1.139} = \frac{5}{1.139} \approx 4.39$

7. Now, we find the probability corresponding to these z-scores. Using the z-tables (or normal distribution calculator):

   • The probability corresponding to $z_{45} = -4.39$ is very close to 0.
   • The probability corresponding to $z_{55} = 4.39$ is very close to 1.

8. The probability that the sample mean is between 45 and 55 is approximately: $P(45 < \bar{x} < 55) = P(z_{45} < z < z_{55}) \approx 1 - 0 = 1$

Conclusion:

The probability that the sample mean is between 45 and 55 is approximately 1 (or very close to 1).

Would you like more details, or do you have any questions?

Here are 5 related questions you might be interested in:

1. How would the probability change if we increase the sample size?
2. What if the population standard deviation was different?
3. How does the Central Limit Theorem help in this case?
4. What is the probability the sample mean is below 48?
5. Can we calculate this for a non-normal population distribution?

Tip: The standard error of the mean decreases as the sample size increases, making the sample mean more precise as a representation of the population mean.