Two hundred thirty-two of the 358 high school students surveyed at a local school say they plan on taking some form of postsecondary education. Calculate the margin of error (ME), rounded to the nearest tenth of a percent. Is it reasonable that the state education department claims the percentage for the entire state is 68%? Justify your answer.
Hint: [tex]ME= \pm 2 \cdot \sqrt{\frac{\hat{p}(1-\hat{p})}{N}}[/tex], where [tex]$\hat{p}$[/tex] is the yes proportion in the sample and [tex]$N$[/tex] is the sample size
A. No, it is not reasonable because the ME is 5.0% and the population proportion of 65% falls outside the sample interval of 73.3% \pm 5.0%
B. Yes, it is reasonable because the ME is 5.0% and the population proportion of 68% falls within the sample interval of 64.8% \pm 5.0%
C. Yes, it is reasonable because the ME is 6.3% and the population proportion of 68% falls within the sample interval of 64.8% \pm 6.3%
D. No, it is not reasonable because the Me is 2.5% and the population proportion of 68% falls outside the sample interval of 64.8% \pm 2.5%
Answer (1)
Calculate the sample proportion: p ^ = 358 232 ≈ 0.648 .
Calculate the margin of error: ME = 2 ⋅ 358 0.648 ( 1 − 0.648 ) ≈ 0.050 .
Convert the margin of error to a percentage: ME ≈ 5.0% .
Since 68% falls within the interval 64.8% ± 5.0% , it is reasonable. Y es , i t i sre a so nab l e b ec a u se t h e ME i s 5.0% an d t h e p o p u l a t i o n p ro p or t i o n o f 68% f a ll s w i t hin t h es am pl e in t er v a l o f 64.8% ± 5.0%
Explanation
Calculate the sample proportion First, we need to calculate the sample proportion, denoted as p ^ . This is the number of students planning postsecondary education divided by the total number of students surveyed.
Calculate sample proportion Given that 232 out of 358 students plan on taking some form of postsecondary education, we calculate the sample proportion as follows: p ^ = 358 232 ≈ 0.648
State the formula for Margin of Error Next, we calculate the margin of error (ME) using the formula: ME = ± 2 ⋅ N p ^ ( 1 − p ^ ) where p ^ is the sample proportion and N is the sample size.
Calculate Margin of Error Plugging in the values, we get: ME = 2 ⋅ 358 0.648 ( 1 − 0.648 ) ME = 2 ⋅ 358 0.648 ( 0.352 ) ME = 2 ⋅ 358 0.228096 ME = 2 ⋅ 0.00063742 ME = 2 ⋅ 0.025247 ME ≈ 0.0505
Convert to percentage and round Converting this to a percentage and rounding to the nearest tenth of a percent, we get: ME ≈ 0.0505 × 100 = 5.05% ME ≈ 5.0%
Calculate the interval Now, we calculate the interval by adding and subtracting the margin of error from the sample proportion: Lower bound: p ^ − ME = 0.648 − 0.0505 = 0.5975 ≈ 59.8% Upper bound: p ^ + ME = 0.648 + 0.0505 = 0.6985 ≈ 69.9%
Check if the claim is reasonable The interval is approximately 59.8% to 69.9% . The state education department claims the percentage for the entire state is 68% . Since 68% falls within the interval 59.8% to 69.9% , it is reasonable.
Final Answer and Justification Therefore, the margin of error is approximately 5.0% , and it is reasonable that the state education department claims the percentage for the entire state is 68% because 68% falls within the sample interval of 64.8% ± 5.0% .
Examples
In market research, understanding the margin of error is crucial when surveying a sample of customers to infer the preferences of the entire customer base. For example, if a survey indicates that 60% of customers prefer a new product with a margin of error of 5%, it suggests the true percentage of customers who prefer the product lies between 55% and 65%. This information helps companies make informed decisions about product launches, marketing strategies, and inventory management, ensuring they align with the actual customer preferences within a reasonable range of certainty. By understanding the margin of error, businesses can avoid making decisions based on potentially misleading sample data.
