Mr. Thomas of Wilmer Amina Carter High School wants to …

Mathematics Questions
Short Answer
Step by Step Solution
Mathematics Questions

Mr. Thomas of Wilmer Amina Carter High School (a large urban high school) wants to know what proportion of the student body favors banning plastic water bottles from the school buildings and grounds. A simple random sample of 152 students finds that 85 support banning plastic bottles. (a) Construct and interpret a 95% confidence interval for the proportion of all students at Wilmer Amina Carter High School who support banning plastic water bottles. State: Plan: Do: Conclude: (b) The student council will ban the bottles if they are convinced that the majority of students favor it. Does this confidence interval provide evidence for a ban? Explain. (c) It turns out that the council’s simple random sample was originally 165 students, but 13 individuals in the sample didn’t respond because they were on an Environmental Science field trip. Could this change your answer to part (b)? Explain your reasoning with a calculation. (d) In part (a) you constructed a 95% confidence interval for the proportion of students at Wilmer Amina Carter High School who support banning plastic water bottles. Explain what you would expect to happen to the length of the interval if the sample size was doubled.

Short Answer

A confidence interval (CI) estimates the unknown proportion of Wilmer High School students supporting a ban on plastic bottles, with a 95% CI calculated as (0.4803, 0.6381). This suggests that between 48% and 63% of students may oppose the ban, indicating the student council should reconsider enforcing it based on this evidence.

Step-by-Step Solution

Step 1: Understand Confidence Interval

A confidence interval (CI) is a range of values used to estimate an unknown population parameter. In this case, we are determining a 95% confidence interval for the percentage of Wilmer High School students who support banning plastic bottles. This interval gives us an idea of the uncertainty surrounding the sample proportion based on a selected level of confidence, which in this situation is 95%.

Step 2: Calculate the Confidence Interval

To compute the 95% confidence interval, use the formula: CI = (hat{p} pm Z sqrt{frac{hat{p}(1-hat{p})}{n}}), where:

  • (hat{p}) = sample proportion (0.5592)
  • n = sample size (152)
  • Z = critical value for 95% confidence (1.96)

Substituting these values into the formula gives us a CI of approximately (0.4803, 0.6381), indicating that between 48% and 63% of students potentially support the ban.

Step 3: Interpret and Conclude the Results

The results of the confidence interval imply that if we were to take many samples and repeat this process, 95% of such intervals would contain the true proportion of students favoring the ban. Since this interval ranges from 48% to 63%, it suggests that a majority of students may not support the ban definitively. Thus, the student council is advised not to outlaw the use of plastic bottles based on this evidence.

Related Concepts

Confidence Interval

A range of values used to estimate an unknown population parameter, indicating the uncertainty surrounding a sample proportion based on a selected level of confidence.

Sample Proportion

The proportion of a certain characteristic (e.g., support for banning plastic bottles) within a selected sample, used to estimate what the true proportion may be in the entire population.

Critical Value

A statistical value that represents the number of standard deviations a data point is from the mean, used in the context of confidence intervals to determine the margin of error for a specific level of confidence.

Table Of Contents
  1. Mr. Thomas of Wilmer Amina Carter High School (a large urban high school) wants to know what proportion of the student body favors banning plastic water bottles from the school buildings and grounds. A simple random sample of 152 students finds that 85 support banning plastic bottles. (a) Construct and interpret a 95% confidence interval for the proportion of all students at Wilmer Amina Carter High School who support banning plastic water bottles. State: Plan: Do: Conclude: (b) The student council will ban the bottles if they are convinced that the majority of students favor it. Does this confidence interval provide evidence for a ban? Explain. (c) It turns out that the council's simple random sample was originally 165 students, but 13 individuals in the sample didn't respond because they were on an Environmental Science field trip. Could this change your answer to part (b)? Explain your reasoning with a calculation. (d) In part (a) you constructed a 95% confidence interval for the proportion of students at Wilmer Amina Carter High School who support banning plastic water bottles. Explain what you would expect to happen to the length of the interval if the sample size was doubled.

Related Articles

Scroll to Top