Suppose x has a mound-shaped distribution with sigma = 9 . A random sample of size 36 has sample mean 20. Is it appropriate to use a normal distribution to compute a confidence interval for the population mean m? Explain. Find a 95% confidence interval for m. Explain the meaning of the confidence interval you computed.
95% = mean ± 1.96 SEm
SEm = SD/√n
I'll let you explain what it means.
To determine if it is appropriate to use a normal distribution to compute a confidence interval for the population mean (m), we need to check if the given conditions are satisfied:
1. Sample size: A general guideline is that the sample size should be sufficiently large (usually greater than 30) to ensure the sampling distribution of the mean is approximately normal. In this case, the sample size is 36, which satisfies this condition.
2. Population distribution: The question mentions that the distribution of x is mound-shaped, which indicates a normal distribution. Therefore, we can assume that the population distribution is normal.
3. Independence: The observations in the sample should be independent of each other. This condition should be met during the random sampling process, assuming the sample was selected randomly.
Since the conditions are satisfied, it is appropriate to use a normal distribution to compute a confidence interval for the population mean.
To find a 95% confidence interval for m, we can use the formula:
Confidence Interval = sample mean ± (Z * standard error)
Here, Z is the critical value that corresponds to the desired confidence level (95% in this case) and the standard error is given by the formula:
Standard Error (SE) = sigma / sqrt(n)
where sigma is the known population standard deviation and n is the sample size.
Substituting the given values, we have:
sigma = 9
sample mean = 20
sample size (n) = 36
Calculating the standard error:
SE = 9 / sqrt(36) = 9 / 6 = 1.5
The critical value for a 95% confidence level is approximately 1.96 (based on a standard normal distribution).
Now, we can calculate the confidence interval:
Confidence Interval = 20 ± (1.96 * 1.5)
Lower limit = 20 - (1.96 * 1.5) = 20 - 2.94 = 17.06
Upper limit = 20 + (1.96 * 1.5) = 20 + 2.94 = 22.94
Therefore, the 95% confidence interval for m is (17.06, 22.94).
The meaning of this confidence interval is that we can be 95% confident that the population mean (m) falls within the range of 17.06 to 22.94. In other words, if we were to repeatedly take random samples and compute confidence intervals using this method, we would expect that around 95% of those intervals would contain the true population mean.
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 use of the normal distribution are met. These conditions are as follows:
1. Randomness: The sample should be selected randomly from the population.
2. Independence: Each observation in the sample should be independent of each other.
3. Sample Size: The sample size should be large enough (approximately, n ≥ 30) or if the population distribution is known to be approximately normal.
In this case, we have a random sample of size 36, so the sample size criterion is met. However, we don't have any information about the population distribution, so we can't directly confirm if the mound-shaped distribution of x is approximately normal. Hence, we need to assume that the population distribution is approximately normal based on the central limit theorem.
The central limit theorem states that for a sufficiently large sample size, the sampling distribution of the sample mean tends to follow a normal distribution, regardless of the shape of the population distribution. Therefore, we can use a normal distribution to compute a confidence interval for the population mean, given that the sample size is large enough.
Now, to find a 95% confidence interval for the population mean, we can use the following formula:
Confidence Interval = Sample mean ± (Critical value) * (Standard error)
The critical value corresponds to the desired level of confidence (95% in this case) and can be obtained from a standard normal distribution table or using statistical software. The standard error is the standard deviation of the sampling distribution, which in this case is the population standard deviation (sigma) divided by the square root of the sample size (sqrt(n)).
Since we know that sigma = 9 and the sample mean = 20, we can plug these values into the formula to calculate the confidence interval.
Standard error = 9 / sqrt(36) = 9 / 6 = 1.5
The critical value for a 95% confidence interval is approximately 1.96.
Confidence Interval = 20 ± 1.96 * 1.5 = (17.62, 22.38)
The confidence interval of (17.62, 22.38) means that we are 95% confident that the true population mean falls within this range. In other words, if we were to take multiple random samples of the same size from the population and calculate a confidence interval for each sample, about 95% of those intervals would contain the true population mean.