Officials are interested to know the average score of all Australian students taking a new English test. How large is the random sample if they want a 90% confidence that the error will not be more than 2 points? It is known that the minimum score and maximum score are 30 and 97 points, respectively.
Question
Officials are interested to know the average score of all Australian students taking a new English test. How large is the random sample if they want a 90% confidence that the error will not be more than 2 points? It is known that the minimum score and maximum score are 30 and 97 points, respectively.
Solution
To answer this question, we need to use the formula for sample size in statistics. The formula is:
n = (Z^2 * σ^2) / E^2
Where:
- n is the sample size
- Z is the Z-value from the Z-table which corresponds to the desired confidence level
- σ is the standard deviation of the population
- E is the desired margin of error
Given that we want a 90% confidence level, the Z-value is 1.645 (you can find this value in a Z-table).
The standard deviation (σ) can be estimated from the range (max - min) by dividing by 4. This is a rule of thumb for estimating standard deviation when it's not provided. So, σ = (97 - 30) / 4 = 16.75
The desired margin of error (E) is 2 points.
Substituting these values into the formula gives:
n = (1.645^2 * 16.75^2) / 2^2 n = (2.706025 * 280.5625) / 4 n = 758.202 / 4 n = 189.55
Since we can't have a fraction of a student, we round up to the nearest whole number. So, the officials would need a sample size of 190 students to be 90% confident that the error will not be more than 2 points.
Similar Questions
Scores on a recent Statistics test are normally distributed with a standard deviation of 6.5 points. If the professor wants to estimate the population mean test score within 1 point with 90% confidence, what sample size is needed
Australia now enter the furthest stage of the tournament in the football team's history and will go into the game against England in the semi-finals of the FIFA Women's World Cup 2023. A survey of 1000 randomly selected Australian football fans are asked if Australia can beat England in the semi-finals. 76% of them believe that Australia can beat England in the semi-finals. Construct a 95% confidence interval around the proportion of Australian football fans who believe that Australia can beat England in the semi-finals. What is the lower bound of this interval (round your answer to four decimal places) ?
Question 4Your instructor tells you that the recent exam had a mean of 80 points and a standard deviation of 2.3. Interpret the standard deviation value in everyday language.1 pointThe maximum score was 82.3.Most students earned an 80 on the exam.The typical score was 2.3 points away from the mean.The range of scores was 4.6.
Question 22You’re going to draw a random sample of professional football players because you want to know what percentage have completed high school. You want to have a margin of error of up to 0.03 at a confidence level of 90%. How big should your sample be?1 pointMinimum 456At least 30Minimum 748Minimum 1068
The standard deviation of test scores on a certain achievement test is 11.3. A random sample of 50 scores on this test had a mean of 75.9. Based on this sample, find a 95% confidence interval for the true mean of all scores. Then give its lower limit and upper limit.Carry your intermediate computations to at least three decimal places. Round your answers to one decimal place. (If necessary, consult a list of formulas.)Lower limit: Upper limit:
Upgrade your grade with Knowee
Get personalized homework help. Review tough concepts in more detail, or go deeper into your topic by exploring other relevant questions.