The National Center for Education Statistics reported 4th-grade math scores in 2003 to have mean 234 and standard deviation 85 (where possible scores range from 0 to 500).
a. Suppose that in a sample of 3,154 4th graders, the mean math score was 228. Standardize to find the z-score for this sample mean.
b. Use your z-score and the sketch of the 68- 95-99.7 Rule (shown on the top in the preceding exercise) to estimate the probability that a random sample of size 3,154 from this population would produce a sample mean of 228 or lower (choose one of the following): greater than 0.16, between 0.16 and 0.025, between 0.025 and 0.0015, or less than 0.0015.
c. Use your z-score and the sketch of the tails of the normal curve (shown on the bottom in the preceding exercise) to estimate the probability that a random sample of size 3,154 from this population
would produce a sample mean of 228 or lower (choose one of the following): greater than 0.05, between 0.05 and 0.025, between 0.025 and 0.01, between 0.01 and 0.005, or less than 0.005.
d. Based on your estimate for the probability of a sample mean equal to 228 or lower when a random sample of size 3,154 is taken from a population whose mean is 234 and standard deviation is 85, characterize such a sample mean as being not improbable, very unlikely, or extremely unlikely.
e. In fact, the sample of 3,154 4th graders were all charter school students. Which of the following is supported by the sample data?
1. The mean math score for charter school 4th graders is fairly close to the mean score for all 4th graders.
2. The mean math score for charter school 4th graders is unusually low compared to the mean score for all 4th graders.
3. The mean math score for charter school 4th graders is unusually high compared to the mean score for all 4th graders.
f. Suppose a sample of just 25 charter schools had a mean score of 228. Find the z-score in this case.
g. Based on the z-score you found in part (f), characterize a sample mean of 228 from a sample of size 25 as being fairly close to 234, unusually low compared to 234, or unusually high compared to 234.