Suppose X has a mound-shaped distribution with sigma=9. A random sample of size 36 has a sample mean 20. Is it appropriate to use a normal distribution to compute a confidence interval for the population mean u?
As long as the mound approximates a normal distribution, yes.
Yes, when the sample size is larger than 30, you assume a normal distribution instead of using the student's t distribution.
To determine if it is appropriate to use a normal distribution to compute a confidence interval for the population mean (u), we need to check if the sample size is large enough and if the population distribution is approximately normal.
In this case, the sample size is 36, which is considered large. The general rule of thumb is that if the sample size (n) is greater than or equal to 30, it is generally acceptable to use a normal distribution.
However, we also need to consider the shape of the population distribution. You mentioned that X has a mound-shaped distribution, which is another name for a normal distribution. This indicates that the population distribution is likely to be approximately normal.
So, with a large sample size (n = 36) and a mound-shaped distribution, it is indeed appropriate to use a normal distribution to compute a confidence interval for the population mean (u).
To determine if it is appropriate to use a normal distribution to compute a confidence interval for the population mean, we need to check if the conditions for the Central Limit Theorem (CLT) are satisfied.
The Central Limit Theorem states that for a sufficiently large sample size, the sample mean will follow a normal distribution, regardless of the shape of the population distribution.
The conditions for the CLT are as follows:
1. Random Sample: The data should be collected through a random sampling process to ensure representative results.
2. Independence: Each observation in the sample should be independent of each other. This means that the value of one observation should not influence the value of another.
3. Sample Size: The sample size should be large enough. While there is no fixed rule for what constitutes a "sufficiently large" sample size, a commonly accepted guideline is that the sample size should be greater than or equal to 30.
In this case, we are given that we have a random sample of size 36. Therefore, the sample size satisfies the condition for the CLT.
Since we don't have information about the sampling process or the independence of the observations, we cannot determine if the first two conditions are met.
Given that the sample size is large enough (36), it can be assumed that the sample mean follows an approximately normal distribution, even if the population distribution is not normally distributed. Therefore, it is appropriate to use a normal distribution to compute a confidence interval for the population mean.
To obtain the confidence interval, you will need to use the formula:
CI = sample mean ± Z * (sigma / sqrt(n))
where:
- sample mean is the given value (20 in this case)
- Z is the critical value from the standard normal distribution corresponding to the desired level of confidence
- sigma is the population standard deviation (given as 9 in this case)
- n is the sample size (given as 36 in this case)
By plugging in the values, you can calculate the confidence interval for the population mean.