A researcher wishes to estimate, with 9999% confidence, the population proportion of adults who think Congress is doing a goo
d or excellent job. Her estimate must be accurate within 44% of the true proportion. (a) No preliminary estimate is available. Find the minimum sample size needed. (b) Find the minimum sample size needed, using a prior study that found that 2222% of the respondents said they think Congress is doing a good or excellent job. (c) Compare the results from parts (a) and (b). (a) What is the minimum sample size needed assuming that no prior information is available? nequals=nothing (Round up to the nearest whole number as needed.)
1 answer:
Answer:
a: 1037 is the minimum sample size needed
b: 712 is the minimum sample size needed
Step-by-step explanation:
We need to use the formula for minimum sample size of a proportion when a sample proportion is known.
The level of confidence is 99%, which has a corresponding z-value of 2.575.
We know the desired error is 4%, or 0.04.
Part a: We have no prior estimate. See attached photo for calculation
Part b:
We know p-hat = 0.22. Therefore q-hat = 1 - 0.22 = 0.78
See the attached photo for the calculation of the minimum sample size
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<span>Let's analyze Hannah's work, step-by-step, to see if she made any mistakes.
</span>In Step 1, Hannah wrote
<span> as the sum of two separate derivatives </span>
<span>using the </span><span>sum rule.
</span>
This step is perfectly fine.
In Step 2,
was kept as it is, and 
was rewritten as 
using the constant rule.Indeed, according to the constant rule, the derivative of a constant number is equal to zero.
This step is perfectly fine.
In Step 3,
was rewritten as 
supposedly using the constant multiple rule.
The problem is that according to the constant multiple rule,
should be rewritten as 
and not as 
.
<span>Therefore, Hannah made a mistake in this step.</span>
</span>In Step 1, Hannah wrote
</span>
This step is perfectly fine.
In Step 2,
This step is perfectly fine.
In Step 3,
The problem is that according to the constant multiple rule,
<span>Therefore, Hannah made a mistake in this step.</span>