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Chapter 7: Problem 12

Blue M\&Ms Express the confidence interval \(0.270 \pm 0.073\) in the form of \(\hat{p}-E

Short Answer

Expert verified
The confidence interval is \[ 0.197 < p < 0.343 \].

Step by step solution

01

Identify \( \hat{p} \) and \( E \)

The given confidence interval is \( 0.270 \pm 0.073 \). Here, \( \hat{p} \) represents the sample proportion, and \( E \) represents the margin of error.
02

Assign values

From the given information, \( \hat{p} = 0.270 \) and \( E = 0.073 \).
03

Write the confidence interval

Using the form \( \hat{p} - E < p < \hat{p} + E \), substitute the values to get: \[ 0.270 - 0.073 < p < 0.270 + 0.073 \].
04

Simplify

Perform the arithmetic operations: \[ 0.197 < p < 0.343 \].

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Key Concepts

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

sample proportion
When dealing with statistics, understanding sample proportion is crucial.
The sample proportion, denoted as \( \hat{p} \), is the ratio of a specific characteristic in a sample to the total number of observations. This value represents an estimate of the true proportion in the population.
For example, if you're estimating the proportion of blue M&M's in a large bag (the population) and you take a sample of 100 M&M's and find 27 are blue, then the sample proportion \( \hat{p} \) is \frac{27}{100} = 0.27.\
The sample proportion is important because it allows us to make inferences about the population from which the sample was drawn.
  • It's a key part of calculating confidence intervals and hypothesis tests.
  • Sample size matters: Larger samples lead to more accurate estimates.
margin of error
The margin of error (E) quantifies the uncertainty in the estimate of the sample proportion. It shows how much the sample proportion might differ from the true population proportion.
In the context of confidence intervals, the margin of error indicates the range around the sample proportion that is likely to contain the true population proportion.
In our example, with a confidence interval of \(0.270 \pm 0.073\), the margin of error is \(0.073\).
The margin of error depends on several factors:
  • The confidence level: Higher confidence levels (like 95% or 99%) lead to larger margins of error.
  • The sample size: Larger samples result in smaller margins of error.
  • The variability in the population: More variability results in larger margins of error.
confidence interval calculation
Constructing a confidence interval allows us to estimate the true population proportion with a certain level of confidence.
The formula for a confidence interval is: \[ \hat{p} - E < p < \hat{p} + E \] We substitute the sample proportion and the margin of error into this formula.
In our example:
  • \hat{p} = 0.270
  • E = 0.073
Using these values, the confidence interval is: \[ 0.270 - 0.073 < p < 0.270 + 0.073 = 0.197 < p < 0.343 \]
This means we are confident that the true population proportion of blue M&M's is between 0.197 and 0.343.
Confidence intervals provide valuable insights:
  • They give a range that is likely to contain the population parameter.
  • They reflect the precision of our estimate.
  • Smaller intervals indicate more precise estimates.

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Most popular questions from this chapter

When she was 9 years of age, Emily Rosa did a science fair experiment in which she tested professional touch therapists to see if they could sense her energy field. She flipped a coin to select either her right hand or her left hand, and then she asked the therapists to identify the selected hand by placing their hand just under Emily's hand without seeing it and without touching it. Among 280 trials, the touch therapists were correct 123 times (based on data in "A Close Look at Therapeutic Touch," Journal of the American Medical Association, Vol. 279, No. 13 ). a. Given that Emily used a coin toss to select either her right hand or her left hand, what proportion of correct responses would be expected if the touch therapists made random guesses? b. Using Emily's sample results, what is the best point estimate of the therapists' success rate? c. Using Emily's sample results, construct a \(99 \%\) confidence interval estimate of the proportion of correct responses made by touch therapists. d. What do the results suggest about the ability of touch therapists to select the correct hand by sensing an energy field?

In a study of speed dating conducted at Columbia University, female subjects were asked to rate the attractiveness of their male dates, and a sample of the results is listed below \((1=\) not attractive; \(10=\) extremely attractive). Construct a \(95 \%\) confidence interval estimate of the standard deviation of the population from which the sample was obtained.

What is different about the normality requirement for a confidence interval estimate of \(\sigma\) and the normality requirement for a confidence interval estimate of \(\mu\) ?

Here is a sample of measured radiation emissions \((\mathrm{cW} / \mathrm{kg})\) for cell phones (based on data from the Environmental Working Group): \(38,55,86,145\). Here are ten bootstrapsamples: \(\\{38,145,55,86\\},\\{86,38,145,145\\},\\{145,86,55,55\\},\\{55,55,55,145\\}\), \(\\{86,86,55,55\\},\\{38,38,86,86\\},\\{145,38,86,55\\},\\{55,86,86,86\\},\\{145,86,55,86\\}\), \(\\{38,145,86,55\\}\) a. Using only the ten given bootstrap samples, construct an \(80 \%\) confidence interval estimate of the population mean. b. Using only the ten given bootstrap samples, construct an \(80 \%\) confidence interval estimate of the population standard deviation.

Find the critical value \(z_{\alpha / 2}\) that corresponds to the given confidence level. \(99.5 \%\)
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