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Chapter 8: Problem 5

Briefly explain how the width of a confidence interval decreases with an increase in the sample size. Give an example.

Short Answer

Expert verified
The width of a confidence interval tends to decrease with an increase in sample size because a larger sample provides more information about the population, thereby reducing the amount of uncertainty. In the given example, an increase in the sample size from 10 to 100 students reduced the confidence interval from 卤15 cm to 卤5 cm, pointing out the less uncertainty about our estimate of the average height.

Step by step solution

01

Understanding Confidence Interval

A confidence interval is a range of values, derived from a statistical procedure, that is likely to contain the value of an unknown population parameter. It provides an interval estimate that describes the uncertainty and variability surrounding the true population parameter.
02

Relationship between Confidence Interval and Sample Size

The width of a confidence interval gives us an idea of how uncertain we are about our sample estimate. If the width of the confidence interval is smaller, it means we have less uncertainty about our estimate. Conversely, a larger confidence interval indicates greater uncertainty. When the sample size increases, we can gain more precise knowledge about our population, which can decrease the width of the confidence interval.
03

Giving an Example

Let's consider an example where you are estimating the average height of students in a school, based on a sample of students. If you take a sample of 10 students, you might find an average height of 170 cm with a confidence interval of 卤15 cm. This means we are 95% confident that the population mean is between 155 cm and 185 cm. Now, suppose you increase the sample size to 100 students. You might find that the average height is still around 170 cm, but the confidence interval might decrease to 卤5 cm. So, we are now 95% confident that the population mean is between 165 cm and 175 cm. The increase in sample size has thus reduced the uncertainty in our estimate of the average height.

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Most popular questions from this chapter

For a population, the value of the standard deviation is \(2.65\). A random sample of 35 observations taken from this population produced the following data. \(\begin{array}{lllllll}42 & 51 & 42 & 31 & 28 & 36 & 49 \\ 29 & 46 & 37 & 32 & 27 & 33 & 41 \\ 47 & 41 & 28 & 46 & 34 & 39 & 48 \\ 26 & 35 & 37 & 38 & 46 & 48 & 39 \\ 29 & 31 & 44 & 41 & 37 & 38 & 46\end{array}\) a. What is the point estimate of \(\mu\) ? b. Make a \(98 \%\) confidence interval for \(\mu\). c. What is the margin of error of estimate for part b?

The standard deviation for a population is \(\sigma=14.8\). A random sample of 25 observations selected from this population gave a mean equal to \(143.72\). The population is known to have a normal distribution. a. Make a \(99 \%\) confidence interval for \(\mu\). b. Construct a \(95 \%\) confidence interval for \(\mu\). c. Determine a \(90 \%\) confidence interval for \(\mu\). d. Does the width of the confidence intervals constructed in parts a through \(\mathrm{c}\) decrease as the confidence level decreases? Explain your answer.

A bank manager wants to know the mean amount of mortgage paid per month by homeowners in an area. A random sample of 120 homeowners selected from this area showed that they pay an average of \(\$ 1575\) per month for their mortgages. The population standard deviation of all such mortgages is \(\$ 215\). a. Find a \(97 \%\) confidence interval for the mean amount of mortgage paid per month by all homeowners in this area. b. Suppose the confidence interval obtained in part a is too wide. How can the width of this interval be reduced? Discuss all possible alternatives. Which alternative is the best?

A city planner wants to estimate the average monthly residential water usage in the city. He selected a random sample of 40 households from the city, which gave a mean water usage of \(3415.70\) gallons over a 1-month period. Based on earlier data, the population standard deviation of the monthly residential water usage in this city is \(389.60\) gallons. Make a \(95 \%\) confidence interval for the average monthly residential water usage for all households in this city.

A large city with chronic economic problems is considering legalizing casino gambling. The city council wants to estimate the proportion of all adults in the city who favor legalized casino gambling. Assume that a preliminary sample has shown that \(63 \%\) of the adults in this city favor legalized casino gambling. How large should the sample size be so that the \(95 \%\) confidence interval for the population proportion has a margin of error of \(.05 ?\)
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