Assume that a test is given to a large number of people but we do not yet know their scores or the shape of the score distribution. Can we be sure that the sampling distribution of the mean for this test will be normally distributed? Why or why not?
The larger the sample size, the more you can safely approximate the normal distribution.
To determine whether the sampling distribution of the mean for a test will be normally distributed, we need to consider the Central Limit Theorem (CLT). The CLT states that, for a sample size large enough, the sampling distribution of the mean tends to follow a normal distribution, regardless of the shape of the underlying population distribution.
Therefore, even if we do not yet know the scores or the shape of the score distribution for the test, if the sample size is sufficiently large, we can reasonably assume that the sampling distribution of the mean will be approximately normally distributed.
The Central Limit Theorem holds under the following conditions:
1. The sample must be random, meaning that every individual has an equal chance of being selected.
2. The sample size should be large enough, typically greater than 30. However, in some cases, even smaller sample sizes may be acceptable, depending on the nature of the population distribution.
3. The observations within the sample should be independent of each other. This means that one individual's score should not be influenced by another individual's score.
By satisfying these conditions, the CLT allows us to rely on the assumption of a normal distribution for the sampling distribution of the mean, even if we do not know the scores or the shape of the score distribution initially.