Let's examine each statement in detail to determine if it is true or false, addressing the implications of the central limit theorem.
1. Statement: If the population is normally distributed, then the mean of a sample from that population will be normally distributed.
- This statement is true. If a population is normally distributed, the sample means drawn from that population will also be normally distributed regardless of the sample size.
2. Statement: For an infinite number of samples, [tex]$99.7 \%$[/tex] of the sample means would fall within the interval [tex]$\mu \pm \frac{\sigma}{\sqrt{n}}$[/tex].
- This statement is false. For an infinite number of samples, 99.7% of the sample means would fall within the interval [tex]$\mu \pm 3 \cdot \frac{\sigma}{\sqrt{n}}$[/tex], not [tex]$\mu \pm \frac{\sigma}{\sqrt{n}}$[/tex].
3. Statement: The mean of the population is the same as the mean of any sample taken from the population.
- This statement is false. The mean of any given sample can deviate from the population mean due to sampling error. While the expected value of the sample mean equals the population mean, individual samples may vary.
4. Statement: For an infinite number of samples, [tex]$95\%$[/tex] of the sample means would fall within the interval [tex]$\mu \pm 2 \cdot \frac{\sigma}{\sqrt{n}}$[/tex].
- This statement is true. For an infinite number of samples, 95% of the sample means would fall within the interval [tex]$\mu \pm 1.96 \cdot \frac{\sigma}{\sqrt{n}}$[/tex], which is often approximated to [tex]$\mu \pm 2 \cdot \frac{\sigma}{\sqrt{n}}$[/tex].
5. Statement: As the number of sample means decreases, the means get closer to a standard normal distribution.
- This statement is false. The central limit theorem indicates that as the number of samples increases, the distribution of the sample means approaches a normal distribution.
6. Statement: The central limit theorem holds true only for populations that are normally distributed.
- This statement is false. The central limit theorem holds true for any population distribution, not just normal distributions. It states that the distribution of the sample means will approximate a normal distribution as the sample size grows, regardless of the original population distribution.
### Conclusion:
The correct statements regarding the implications of the central limit theorem are:
- Statement 1: If the population is normally distributed, then the mean of a sample from that population will be normally distributed.
- Statement 4: For an infinite number of samples, 95% of the sample means would fall within the interval [tex]$\mu \pm 2 \cdot \frac{\sigma}{\sqrt{n}}$[/tex].
Thus, the true statements are numbered 0 and 3, resulting in the answer:
[0, 3]
