1) Washington High School has 100 national merit scholars. You have data from al

Question

1) Washington High School has 100 national merit scholars. You have data from all 100, so you know that the mean IQ across all 100 is 125.3. The principle plans to randomly sample 20 of the national merit scholars and compute a 95% confidence interval with their data.

Lincoln High School also has 100 national merit scholars. You don’t have access to data from all of them, so you don’t know the mean IQ across all 100. However, the principle took a random sample of 20 and computed a 95% confidence interval with their data. This interval ranged from 126.4 to 133.2.

A) In a sentence or two, explain what it means for a value to be included in a confidence interval and what it means for a value to be excluded from a confidence interval.

B) What is the probability that the interval from Washington High School will include the value 125.3, where your options are “5%”, “95%”, and “unknown”?

C) What is the probability that the mean IQ across all 100 national merit scholars at Lincoln High School is somewhere between 126.4 and 133.2, where your options are “5%”, “95%”, and “unknown”?

D) In a sentence or two, explain what alpha means for a confidence interval.

E) Briefly (1-5 sentences) explain how the definition of alpha for a confidence interval relates to your answers for B and C.

Explanation / Answer

Part A

We are 95% confident that the population mean IQ scores will be lies between two values 126.4 and 133.2. We are 5% confident that the population mean IQ scores will not be lies between two values 126.4 and 133.2.

Part B

The probability that the Washington High School will include the value 125.3 is 5%, because the value 125.3 is not lies between the given confidence interval.

Part C

The probability that the mean IQ across all 100 merit scholars at Lincoln High School is somewhere between 126.4 and 133.2 is given as 95%, because confidence level is 95% or we are 95% confident that the population mean IQ scores will be lies between two values 126.4 and 133.2.

Part D

The value of alpha or level of significance is the probability that the population parameter or population mean is not lies between the given confidence interval. The value of confidence level is the probability that the population parameter or population mean is lies between the given confidence interval.

Part E

For part B, the probability of population IQ score will not lies between given confidence interval is 5%, however, for part C, the probability of population IQ score will lies between given confidence interval is 95%. The population for the given scenario is the IQ scores of 100 national merit scholars from Washington High School and sample is given as randomly selected IQ scores of 20 national merit scholars from Washington High School.

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