A recent survey of students on the question “Do you think the United States should retain or abolish the use of the penny as a coin of currency?” showed that 20 of the 28 students thought that the United States should retain the penny. Consider this class as a random sample of all adults.
a) Find a 95% confidence interval for the true proportion that thinks the United States should retain the penny.
b) The Harris Poll® #51, July 15, 2004 reported that almost 60% of are in favor of retaining the penny. Do the class results differ significantly from this poll’s result? Explain.
c) The poll site stated, “In theory, with probability samples of this size (2,136 adults), one could say with 95% certainty that the results have a sampling error of ±2 percentage points of what they would be if the entire adult population had been polled with complete accuracy.” Is this statement correct? Explain.
I have the proportion for the class of 28. How do I figure the same proportion from the 2135 adults in the poll? Also, is the statement correct in c)?
APPRECIATE any guidance. K
To find the same proportion for the 2,136 adults in the poll, you can use a proportion formula. Let's denote the proportion of adults who think the United States should retain the penny as p. And let's denote the sample proportion for the class of 28 students as phat.
Since the class of 28 students is considered a random sample of all adults, we can assume that the proportion of adults who think the United States should retain the penny is the same as the proportion of students in the class. Therefore, we can use the sample proportion for the class, p̂, to estimate the proportion for the entire population of adults, p.
To find p, we can set up a proportion equation:
p̂ = 20/28
p̂ = p/2136
Cross-multiplying, we can solve for p:
p = p̂ * 2136/28
p = (20/28) * 2136/28
p = (0.714) * 76.286
p ≈ 54.49%
So, the proportion for the 2,136 adults in the poll would be approximately 54.49%.
Now, let's move on to the questions a), b), and c).
a) To find a 95% confidence interval for the true proportion that thinks the United States should retain the penny based on the class sample, you can use the formula for a confidence interval:
CI = p̂ ± z * sqrt((p̂(1 - p̂))/n)
Where:
- p̂ is the sample proportion (20/28)
- z is the z-value corresponding to the desired confidence level (95% corresponds to z = 1.96)
- sqrt represents the square root
- n is the sample size (28)
Plugging in the values:
CI = (20/28) ± 1.96 * sqrt((20/28)(1 - 20/28)/28)
CI ≈ 0.714 ± 1.96 * sqrt((0.714)(0.286)/28)
CI ≈ 0.714 ± 0.228
So, the 95% confidence interval for the true proportion of adults who think the United States should retain the penny is approximately 0.486 to 0.942.
b) To determine if the class results differ significantly from the poll result of almost 60%, you can compare the confidence interval for the class result with the poll result.
Since the poll result of 60% is outside the 95% confidence interval of 0.486 to 0.942, we can conclude that the class results differ significantly from the poll result. However, it's important to consider the margin of error and the specific details of the poll methodology to interpret this result accurately.
c) The statement that "with probability samples of size 2,136 adults, one could say with 95% certainty that the results have a sampling error of ±2 percentage points" is correct. A probability sample of this size allows for a specific margin of error, which is ±2 percentage points in this case. This means that if the entire adult population had been surveyed using the same methodology, the results would likely fall within ±2 percentage points of the reported proportions with 95% certainty.
So, the statement in c) is correct.