a guidance counselor at lincoln high school in boston, ma wants to analyze scores on the act to increase the…

a guidance counselor at lincoln high school in boston, ma wants to analyze scores on the act to increase the average scores for seniors. the counselor gathered the act scores for all seniors at the school. the average act score in the us is 20.8. the counselor picked a random sample of 160 senior scores and recorded the information in the table below. use this data to answer questions 1 - 5. 1. what is the proportion of students that scored less than 20.8 on the act at lincoln high school? 2. estimate the probability of a senior in the random sample scoring less than 20.8 on the act? explain your method. 3. there are 580 seniors at lincoln high school. using equivalent fractions, about how many seniors are expected to score less than 20.8 on the act (based on the sample data)? 4. if you use the percent method to answer question 3, what similarities and differences would you notice? explain. 5. the guidance counselor estimates that 312 seniors will score 20.8 or greater on the act sample data sample data. explain whether this would be considered unreasonable based on the sample data.

a guidance counselor at lincoln high school in boston, ma wants to analyze scores on the act to increase the average scores for seniors. the counselor gathered the act scores for all seniors at the school. the average act score in the us is 20.8. the counselor picked a random sample of 160 senior scores and recorded the information in the table below. use this data to answer questions 1 - 5. 1. what is the proportion of students that scored less than 20.8 on the act at lincoln high school? 2. estimate the probability of a senior in the random sample scoring less than 20.8 on the act? explain your method. 3. there are 580 seniors at lincoln high school. using equivalent fractions, about how many seniors are expected to score less than 20.8 on the act (based on the sample data)? 4. if you use the percent method to answer question 3, what similarities and differences would you notice? explain. 5. the guidance counselor estimates that 312 seniors will score 20.8 or greater on the act sample data sample data. explain whether this would be considered unreasonable based on the sample data.

Answer

Explanation:

Step1: Calculate the proportion for question 1

The formula for proportion is (p=\frac{\text{number of favorable outcomes}}{\text{total number of outcomes}}). Here, the number of students who scored less than (20.8) is (104), and the total number of students in the sample is (160). So, (p = \frac{104}{160}). Simplify (\frac{104}{160}) by dividing numerator and denominator by (8): (\frac{104\div8}{160\div8}=\frac{13}{20}=0.65)

Step2: Estimate probability for question 2

Probability is estimated using the sample proportion. Since probability (P) (scoring less than (20.8)) is based on the sample data. The sample proportion (calculated in step 1) is used as an estimate of the probability. So (P=\frac{104}{160} = 0.65)

Step3: Use equivalent fractions for question 3

Let (x) be the number of seniors out of (580) who score less than (20.8). We know that (\frac{104}{160}=\frac{x}{580}) Cross - multiply: (160x=104\times580) (160x = 60320) Solve for (x): (x=\frac{60320}{160}=377)

Answer:

  1. (0.65)
  2. The probability is (0.65). We use the sample proportion (number of students with score less than (20.8) in the sample divided by total sample size) as an estimate of the probability.
  3. (377)
  4. Similarity: Both methods (equivalent fractions and percent) use the ratio of students with score less than (20.8) in the sample. Difference: In equivalent fractions, we set up (\frac{104}{160}=\frac{x}{580}), while in percent method, first find the percentage ((65%)) and then calculate (x = 0.65\times580=377)
  5. First, find the proportion of students who scored (20.8) or higher in the sample: (\frac{56}{160}=0.35). If there are (N = 580) seniors, the expected number of students who score (20.8) or higher is (0.35\times580 = 203). Since (312>203), the counselor's estimate is unreasonable.