"To test H0: mu = 100 versus Ha: mu > 100, a simple random sample of size n = 35 is obtained from an unknown distribution. The sample mean is 104.3 and the sample standard deviation is 12.4.


a. To use the t-distribution, why must the sample size be large?"

Verified step by step guidance

1. Recall that the t-distribution is used when the population standard deviation is unknown and the sample size is small, especially when the population distribution is not normal.
2. Understand that the t-distribution assumes the underlying population is approximately normal, which is important for small samples to ensure the validity of inference.
3. Recognize that when the sample size is large (usually n > 30), the Central Limit Theorem states that the sampling distribution of the sample mean tends to be approximately normal regardless of the population distribution shape.
4. Therefore, a large sample size allows us to use the t-distribution even if the population distribution is unknown or not normal, because the sample mean's distribution will be close to normal.
5. In summary, the sample size must be large to justify the use of the t-distribution without knowing the population distribution, relying on the Central Limit Theorem to approximate normality.

Key Concepts

Central Limit Theorem

The Central Limit Theorem states that the sampling distribution of the sample mean approaches a normal distribution as the sample size becomes large, regardless of the population's original distribution. This allows the use of normal or t-distributions for inference when the sample size is sufficiently large.

t-Distribution and Its Assumptions

The t-distribution is used for hypothesis testing about a population mean when the population standard deviation is unknown and the sample size is small. It assumes the sample comes from a normally distributed population or that the sample size is large enough for the Central Limit Theorem to apply.

Sample Size and Normality

A large sample size (commonly n ≥ 30) helps ensure the sample mean's distribution is approximately normal, even if the population distribution is unknown or not normal. This justifies using the t-distribution for inference, making the test results more reliable.