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Chapter 9: Problem 14

Determine the point estimate of the population proportion, the margin of error for each confidence interval, and the number of individuals in the sample with the specified characteristic, \(x,\) for the sample size provided. Lower bound: \(0.853,\) upper bound: \(0.871, n=10,732\)

Short Answer

Expert verified
Point estimate is 0.862. Margin of error is 0.009. Number of individuals with the characteristic is approximately 9,246.

Step by step solution

01

Determine the Point Estimate of the Population Proportion

The point estimate of the population proportion is the average (or midpoint) of the lower and upper bounds of the confidence interval. Calculate it using the formula: o \[\text{Point Estimate} = \frac{\text{Lower Bound} + \text{Upper Bound}}{2}\]o Substituting the given values: o \[\text{Point Estimate} = \frac{0.853 + 0.871}{2} = 0.862\]
02

Calculate the Margin of Error

The margin of error is half the width of the confidence interval. Calculate it using the formula: o \[\text{Margin of Error} = \frac{\text{Upper Bound} - \text{Lower Bound}}{2}\]o Substituting the given values: o \[\text{Margin of Error} = \frac{0.871 - 0.853}{2} = 0.009\]
03

Determine the Number of Individuals in the Sample with the Specified Characteristic (o\text{x}single-character placeholder)

The number of individuals in the sample with the specified characteristic (o\text{x}single-character placeholder) can be calculated using the point estimate and the sample size. Use the formula: o \[\text{Number of individuals (x)} = \text{Point Estimate} 脳 n\]o Substituting the given values: o \[\text{x} = 0.862 脳 10,732 \approx 9,246\]o Thus: o \[\text{x} \approx 9,246\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Point Estimate
A point estimate is a single value used to estimate a population parameter. In the case of population proportions, we use the midpoint of a confidence interval as the point estimate.
For example, if the lower bound of our confidence interval is 0.853 and the upper bound is 0.871, we can find the point estimate by averaging these two values:
\[\text{Point Estimate} = \frac{\text{Lower Bound} + \text{Upper Bound}}{2} = \frac{0.853 + 0.871}{2} = 0.862\]
So, the point estimate for the population proportion in this case is 0.862. This means we estimate that 86.2% of the population has the specified characteristic.
Margin of Error
The margin of error tells us how much uncertainty there is in our point estimate. It's calculated as half the width of the confidence interval.
Using the previous example with a lower bound of 0.853 and an upper bound of 0.871, the margin of error can be calculated with the formula:
\[\text{Margin of Error} = \frac{\text{Upper Bound} - \text{Lower Bound}}{2} = \frac{0.871 - 0.853}{2} = 0.009\]
This means our point estimate of 0.862 could vary by 卤0.009. So, in the real world, the true population proportion could be as low as 0.853 or as high as 0.871.
Confidence Interval
A confidence interval gives a range within which we believe the true population parameter lies. It is constructed around the point estimate and includes the margin of error.
For example, with a point estimate of 0.862 and a margin of error of 0.009, our confidence interval is:
- Lower bound: \[\text{Point Estimate} - \text{Margin of Error} = 0.862 - 0.009 = 0.853\]
- Upper bound: \[\text{Point Estimate} + \text{Margin of Error} = 0.862 + 0.009 = 0.871\]
So, we are confident that the true population proportion lies between 0.853 and 0.871. The width of the interval reflects the level of confidence or certainty we have about this range.
Sample Size
The sample size impacts the accuracy and precision of our estimates. Larger sample sizes tend to give more accurate results because they reduce the margin of error.
In the previous example, our sample size (n) was 10,732. Given this large sample size, our point estimate and margin of error are likely to be quite accurate.
To find the number of individuals in the sample who have the specified characteristic, we multiply the point estimate by the sample size:
\[\text{Number of individuals (x)} = \text{Point Estimate} \times n = 0.862 \times 10,732 \approx 9,246\]
So, about 9,246 individuals in our sample have the specified characteristic, according to our estimate.

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Most popular questions from this chapter

Determine the critical value \(z_{\alpha / 2}\) that corresponds to the given level of confidence. \(98 \%\)

The trade volume of a stock is the number of shares traded on a given day. The following data, in millions (so that 6.16 represents 6,160,000 shares traded), represent the volume of PepsiCo stock traded for a random sample of 40 trading days in 2014. \begin{array}{llllllll} \hline 6.16 & 6.39 & 5.05 & 4.41 & 4.16 & 4.00 & 2.37 & 7.71 \\ \hline 4.98 & 4.02 & 4.95 & 4.97 & 7.54 & 6.22 & 4.84 & 7.29 \\ \hline 5.55 & 4.35 & 4.42 & 5.07 & 8.88 & 4.64 & 4.13 & 3.94 \\ \hline 4.28 & 6.69 & 3.25 & 4.80 & 7.56 & 6.96 & 6.67 & 5.04 \\ \hline 7.28 & 5.32 & 4.92 & 6.92 & 6.10 & 6.71 & 6.23 & 2.42 \\ \hline \end{array} (a) Use the data to compute a point estimate for the population mean number of shares traded per day in 2014 (b) Construct a \(95 \%\) confidence interval for the population mean number of shares traded per day in 2014 . Interpret the confidence interval. (c) A second random sample of 40 days in 2014 resulted in the data shown next. Construct another \(95 \%\) confidence interval for the population mean number of shares traded per day in 2014\. Interpret the confidence interval. $$ \begin{array}{llllrlll} \hline 6.12 & 5.73 & 6.85 & 5.00 & 4.89 & 3.79 & 5.75 & 6.04 \\ \hline 4.49 & 6.34 & 5.90 & 5.44 & 10.96 & 4.54 & 5.46 & 6.58 \\ \hline 8.57 & 3.65 & 4.52 & 7.76 & 5.27 & 4.85 & 4.81 & 6.74 \\ \hline 3.65 & 4.80 & 3.39 & 5.99 & 7.65 & 8.13 & 6.69 & 4.37 \\ \hline 6.89 & 5.08 & 8.37 & 5.68 & 4.96 & 5.14 & 7.84 & 3.71 \\ \hline \end{array} $$ (d) Explain why the confidence intervals obtained in parts (b) and (c) are different.

The trade magazine QSR routinely checks the drive-through service times of fast-food restaurants. A \(90 \%\) confidence interval that results from examining 607 customers in Taco Bell's drive-through has a lower bound of 161.5 seconds and an upper bound of 164.7 seconds. What does this mean?

In a USA Today/Gallup poll, 768 of 1024 randomly selected adult Americans aged 18 or older stated that a candidate's positions on the issue of family values are extremely or very important in determining their vote for president. (a) Obtain a point estimate for the proportion of adult Americans aged 18 or older for which the issue of family values is extremely or very important in determining their vote for president. (b) Verify that the requirements for constructing a confidence interval for \(p\) are satisfied. (c) Construct a \(99 \%\) confidence interval for the proportion of adult Americans aged 18 or older for which the issue of family values is extremely or very important in determining their vote for president. (d) Is it possible that the proportion of adult Americans aged 18 or older for which the issue of family values is extremely or very important in determining their vote for president is below \(70 \%\) ? Is this likely? (e) Use the results of part (c) to construct a \(99 \%\) confidence interval for the proportion of adult Americans aged 18 or older for which the issue of family values is not extremely or very important in determining their vote for president.

Find the critical values \(\chi_{1-\alpha / 2}^{2}\) and \(\chi_{\alpha / 2}^{2}\) for the given level of confidence and sample size. \(98 \%\) confidence, \(n=23\)
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