stats

How to determine the appropriate tool when the variance is known, variance is unknown, and when central limit theorem is used.
When the variance is known, we can use the z-test to test hypotheses about population means. The formula for the z-test is:z = (x̄ - μ) / (σ / √n)
VARIANCE IS KNOWN
z = (x̄ - μ) / (σ / √n)where x̄ is the sample mean, μ is the population mean, σ is the population standard deviation, and n is the sample size. We use the z-test when the population standard deviation is known.
VARIANCE IS KNOWN
When the variance is unknown, we can use the t-test to test hypotheses about population means. The formula for the t-test is:t = (x̄ - μ) / (s / √n)
VARIANCE IS UNKNOWN
t = (x̄ - μ) / (s / √n)where s is the sample standard deviation. We use the t-test when the population standard deviation is unknown.
VARIANCE IS KNOWN
NOTE:When we have a large sample size (typically n 30), we can use the central limit theorem to approximate the sampling distribution of the sample mean as a normal distribution. In this case, we can use the z-test to test hypotheses about population means, even if the population standard deviation is unknown.
CENTRAL LIMIT THEOREM

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