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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

The principal of a large high school is concerned about the amount of time that his students spend on jobs to pay for their cars, to buy clothes, and so on. He would like to estimate the mean number of hours worked per week by these students. He knows that the standard deviation of the times spent per week on such jobs by all students is \(2.5\) hours. What sample size should he choose so that the estimate is within \(.75\) hour of the population mean? The principal wants to use a \(98 \%\) confidence level.

KidPix Entertainment is in the planning stages of producing a new computer- animated movie for national release, so they need to determine the production time (labor-hours necessary) to produce the movie. The mean production time for a random sample of 14 big-screen computer-animated movies is found to be 53,550 labor-hours. Suppose that the population standard deviation is known to be 7462 labor-hours and the distribution of production times is normal. a. Construct a \(98 \%\) confidence interval for the mean production time to produce a big-screen computer-animated movie. b. Explain why we need to make the confidence interval. Why is it not correct to say that the average production time needed to produce all big-screen computer-animated movies is 53,550 labor-hours?

Briefly explain the meaning of the degrees of freedom for a \(t\) distribution. Give one example.

At Farmer's Dairy, a machine is set to fill 32 -ounce milk cartons. However, this machine does not put exactly 32 ounces of milk into each carton; the amount varies slightly from carton to carton. It is known that when the machine is working properly, the mean net weight of these cartons is 32 ounces. The standard deviation of the amounts of milk in all such cartons is always equal to \(.15\) ounce. The quality control department takes a sample of 25 such cartons every week, calculates the mean net weight of these cartons, and makes a \(99 \%\) confidence interval for the population mean. If either the upper limit of this confidence interval is greater than \(32.15\) ounces or the lower limit of this confidence interval is less than \(31.85\) ounces, the machine is stopped and adjusted. A recent sample of 25 such cartons produced a mean net weight of \(31.94\) ounces. Based on this sample, will you conclude that the machine needs an adjustment? Assume that the amounts of milk put in all such cartons have a normal distribution.

A hospital administration wants to estimate the mean time spent by patients waiting for treatment at the emergency room. The waiting times (in minutes) recorded for a random sample of 35 such patients are given below. \(\begin{array}{lrrrrrr}30 & 7 & 68 & 76 & 47 & 60 & 51 \\ 64 & 25 & 35 & 29 & 30 & 35 & 62 \\ 96 & 104 & 58 & 32 & 32 & 102 & 27 \\ 45 & 11 & 64 & 62 & 72 & 39 & 92 \\ 84 & 47 & 12 & 33 & 55 & 84 & 36\end{array}\) Construct a \(99 \%\) confidence interval for the corresponding population mean. Use the \(t\) distribution.
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