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Chapter 8: Problem 71

Check if the sample size is large enough to use the normal distribution to make a confidence interval for \(p\) for each of the following cases. a. \(n=80\) and \(\hat{p}=.85\) b. \(n=110\) and \(\hat{p}=.98\) c. \(n=35\) and \(\hat{p}=.40\) d. \(n=200\) and \(\hat{p}=.08\)

Short Answer

Expert verified
For these cases, the samples are large enough for A, C, D and not large enough for B

Step by step solution

01

Case A: Checking for \(n=80\) and \(\hat{p}=.85\)

First calculate \(n*\hat{p}\) and \(n*(1-\hat{p})\) for these values. That is \(80*.85 = 68\) and \(80*(1-.85) = 12\). Both these values are greater than 10 and as such, the sample size is large enough for this case.
02

Case B: Checking for \(n=110\) and \(\hat{p}=.98\)

Again, calculate \(n*\hat{p}\) and \(n*(1-\hat{p})\). Here, \(110*.98 = 107.8\) and \(110*(1-.98) = 2.2\). The second value is less than 10. Therefore, for this case the sample size is not large enough.
03

Case C: Checking for \(n=35\) and \(\hat{p}=.40\)

Calculate \(n*\hat{p}\) and \(n*(1-\hat{p})\). For these values, \(35*.40 = 14\) and \(35*(1-.40) = 21\). Both these values are greater than 10, so the sample size is large enough for this case.
04

Case D: Checking for \(n=200\) and \(\hat{p}=.08\)

Again, calculate \(n*\hat{p}\) and \(n*(1-\hat{p})\). That would be: \(200*.08 = 16\) and \(200*(1-.08) = 184\). Both these values are greater than 10, so here the sample size is large enough as well.

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Key Concepts

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

Sample Size
When you want to estimate characteristics of a population, sample size is crucial. A larger sample provides more reliable data.
To decide if a sample size is sufficient for creating a confidence interval using a normal distribution, we apply a rule of thumb:
  • Calculate: \( n \times \hat{p} \) and \( n \times (1 - \hat{p}) \).
  • If both values are greater than 10, the sample size is usually large enough.
This rule helps ensure data is spread out properly to reflect the population. Keep in mind that larger samples reduce error margins and increase confidence.
Remember, the right sample size helps produce meaningful conclusions from your data.
Normal Distribution
The normal distribution is a key concept in statistics, known for its bell-shaped curve.
It's essential when making predictions and conducting statistical analyses. For a dataset, if individual data points are distributed symmetrically around the mean, we often say the data is normally distributed.
  • This is handy for creating confidence intervals because it allows us to make assumptions about the data.
  • Many statistical tests rely on the idea of normal distribution.
When conditions are right, such as having a sufficiently large sample size, we can use the normal distribution to approximate other distributions, like the binomial.
In practical terms, it simplifies complex statistical calculations!
Proportions
Proportions tell you about the relationship of a part to a whole.
It's expressed as a fraction or percentage. For example, \( \hat{p} \) represents the sample proportion.When determining the accuracy of a proportion, consider:
  • Sample proportion \( \hat{p} \) 鈥� the number of successes divided by the total sample size.
  • Understanding \( \hat{p} \) is key when working with confidence intervals because it reflects what happens in your sample.
Proportions are foundational in many fields like business and health to make predictions and decisions based on sampled data.
Therefore, understanding proportions and their estimation is foundational for effective statistical practice.
Statistical Analysis
Statistical analysis is about discovering patterns and relationships in data.
It's the backbone of making informed decisions based on evidence. Here鈥檚 how it works:
  • Collect data 鈥� gather information relevant to your question.
  • Analyze 鈥� use mathematical formulas and methods to describe and infer patterns in your data.
  • Interpret 鈥� make sense of the results to provide insights or support decision-making.
Confidence intervals are part of statistical analysis, offering a range where the true population parameter likely falls.
When conducting statistical analysis, ensure your methods are robust to draw accurate and meaningful conclusions.

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Most popular questions from this chapter

A sample selected from a population gave a sample proportion equal to .31. a. Make a \(95 \%\) confidence interval for \(p\) assuming \(n=1200\). b. Construct a \(95 \%\) confidence interval for \(p\) assuming \(n=500\). c. Make a \(95 \%\) confidence interval for \(p\) assuming \(n=80\). d. Does the width of the confidence intervals constructed in parts a through \(c\) increase as the sample size decreases? If yes, explain why.

According to liposuction4you.com, the maximum amount of fat and fluid that can be removed safely during a liposuction procedure is 6 liters. Suppose that the following data represent the amount of fat and fluid removed during 12 randomly selected liposuction procedures. Assume that the population distribution of such amounts is normal. $$ \begin{array}{llllll} 1.84 & 2.66 & 2.96 & 2.42 & 2.88 & 2.86 \\ 3.66 & 3.65 & 2.33 & 2.66 & 3.20 & 2.24 \end{array} $$ a. What is the point estimate of the corresponding population mean? b. Construct a \(98 \%\) confidence interval for the corresponding population mean.

In June 2008 , SBRI Public Affairs conducted a telephone poll of 1004 adult Americans aged 18 and older. One of the questions asked was, "In the past year, was there ever a time when you ...?" Respondents could choose more than one of the answers mentioned. Of the respondents, \(64 \%\) said "cut back on vacations or entertainment because of their cost," \(37 \%\) said "failed to pay a bill on time," and \(25 \%\) said "have not gone to a doctor because of the cost." (Source: http://www.srbi.com/AmericansConcernEconomic.html.) Using these results, find a \(95 \%\) confidence interval for the corresponding population percentage for each answer. Write a one-page report to present these results to a group of college students who have not taken statistics. Your report should answer questions such as: (1) What is a confidence interval? (2) Why is a range of values more informative than a single percentage? (3) What does \(95 \%\) confidence mean in this context? (4) What assumptions, if any, are you making when you construct each confidence interval?

A sample of 18 observations taken from a normally distributed population produced the following data. \(\begin{array}{lllllllll}28.4 & 27.3 & 25.5 & 25.5 & 31.1 & 23.0 & 26.3 & 24.6 & 28.4\end{array}\) \(\begin{array}{llllllll}37.2 & 23.9 & 28.7 & 27.9 & 25.1 & 27.2 & 25.3\end{array}\) \(\begin{array}{ll}22.6 & 22.7\end{array}\) a. What is the point estimate of \(\mu\) ? b. Make a \(99 \%\) confidence interval for \(\mu .\) c. What is the margin of error of estimate for \(\mu\) in part b?

The U.S. Senate just passed a bill by a vote of \(55-45\) (with all 100 senators voting). A student who took an elementary statistics course last semester says, "We can use these data to make a confidence interval about \(p\). We have \(n=100\) and \(\hat{p}=55 / 100=.55\) " Hence, according to him, a \(95 \%\) confidence interval for \(p\) is $$ \hat{p} \pm z \sigma_{\hat{p}}=.55 \pm 1.96 \sqrt{\frac{(.55)(.45)}{100}}=.55 \pm .098=.452 \text { to } .648 $$ Does this make sense? If not, what is wrong with the student's reasoning?
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