EXAMPLE 4 Is the sampling distribution of normal?

For each of (a)–(c), determine whether the sampling distribution of is normal, approximately normal, or unknown.

  1. Assume SAT Math scores are normally distributed. Samples of size are taken.
  2. Assume SAT Math scores are not normally distributed. Samples of size are taken.
  3. Assume SAT Math scores are not normally distributed. Samples of size are taken.

Solution

  1. Here, the distribution of (SAT Math scores) is normally distributed, so the sampling distribution of is automatically normal (Fact 3, page 399), regardless of the sample size. This is Case 1 from the Three Cases above.
  2. Here, is not normal, but the sample size is considered large because it is greater than 30. Thus, the Central Limit Theorem applies, so that the sampling distribution of is approximately normal. This is Case 2 from the Three Cases above.
  3. Here, we have neither normality nor a large sample size. Therefore, we have insufficient information to conclude that the sampling distribution of the sample mean is either normal or approximately normal. This is Case 3 from the Three Cases above.

NOW YOU CAN DO

Exercises 17–26