The National Assessment of Educational Progress (NAEP) includes a mathematics test for eighth‑grade students. Scores on the test range from 0 to 500 . Demonstrating the ability to use the mean to solve a problem is an example of the skills and knowledge associated with performance at the Basic level. An example of the knowledge and skills associated with the Proficient level is being able to read and interpret a stem‑and‑leaf plot. In 2015, 136,900 eighth‑graders were in the NAEP sample for the mathematics test. The mean mathematics score was ¯=282 . We want to estimate the mean score in the population of all eighth‑graders. Consider the NAEP sample as an SRS from a Normal population with standard deviation =110.

(a) If we take many samples, the sample mean ¯ varies from sample to sample according to a Normal distribution with mean equal to the unknown mean score in the population. What is the standard deviation of this sampling distribution? (Enter your answer rounded to four decimal places.)
(b) According to the 95 part of the 68‑95‑99.7 rule, 95% of all values of ¯ fall within how many points on either side of the unknown mean ? (Enter your answer rounded to four decimal places.)
(c) What is the 95% confidence interval for the population mean score based on this one sample? (Enter your answer rounded to one decimal place.)
Lower limit and upper limit