Please only the formula and step I need for this problem. thank you
In response to media inquiries and concerns expressed by groups opposed to violence, the president of a university with over 25,000 students has agreed to survey a simply random sample of her students to find out whether the student body thinks the school’s “Plundering Pirate” mascot should be changed to one that is less aggressive in name and appearance. Of the 200 students selected for participation in the survey, only 20% believe the school should select a new and more kindly mascot. Construct a 90% confidence interval for the population proportion of students who believe the mascot should be changed. Based on the sample findings and associated confidence interval, comment on the credibility of a local journalist’s comment that “over 50% of the students would like a new mascot.
To construct a confidence interval for the population proportion, you can follow these steps:
Step 1: Calculate the sample proportion.
In this case, the sample proportion is the number of students who believe the mascot should be changed (20% of 200), which is 40 students.
Step 2: Determine the standard error.
The standard error is the estimated standard deviation of the sample proportion and is calculated using the formula:
standard error = √( (p̂ * q̂) / n )
where:
- p̂ is the sample proportion (40/200 = 0.2)
- q̂ is 1 - p̂ (1 - 0.2 = 0.8)
- n is the sample size (200)
standard error = √( (0.2 * 0.8) / 200 ) = 0.027
Step 3: Find the critical value.
To construct a 90% confidence interval, we need to find the critical value associated with a confidence level of 90%. This value can be obtained from a standard normal distribution table or using a calculator. For a two-tailed z-test with a 90% confidence level, the critical value is approximately 1.645.
Step 4: Calculate the margin of error.
The margin of error is determined by multiplying the standard error by the critical value:
margin of error = critical value * standard error
= 1.645 * 0.027 ≈ 0.044
Step 5: Calculate the lower and upper bounds of the confidence interval.
The lower bound of the confidence interval is obtained by subtracting the margin of error from the sample proportion, while the upper bound is obtained by adding the margin of error to the sample proportion:
lower bound = p̂ - margin of error
= 0.2 - 0.044 = 0.156
upper bound = p̂ + margin of error
= 0.2 + 0.044 = 0.244
Therefore, the 90% confidence interval for the population proportion of students who believe the mascot should be changed is approximately 0.156 to 0.244.
Based on this confidence interval, there is no evidence to support the local journalist's comment that "over 50% of the students would like a new mascot." The confidence interval suggests that the true proportion lies between 15.6% and 24.4%, which does not include the claimed proportion of over 50%.