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Chapter 6: Problem 113

What sample size is needed to give the desired margin of error in estimating a population mean with the indicated level of confidence? A margin of error within 卤0.5 with \(90 \%\) confidence, if we make a reasonable estimate that \(\sigma=25\)

Short Answer

Expert verified
The required minimum sample size for a margin of error within 卤0.5 with 90% confidence, given that 蟽=25, is 6727.

Step by step solution

01

Understand Confidence Interval and Required Formula

In statistics, Confidence intervals are used to estimate the range in which our population mean is likely to be found. Based on the requirements of margin of error (E), confidence level and standard deviation (蟽), we need to make use of the following formula which gives us the minimum sample size required: \(n = \left( \frac{Z*\sigma}{E} \right)^2 \). Here, Z is the Z-score corresponding to the given level of confidence (90% in this case) and it can be found from the standard Normal Distribution.
02

Find Z-Score

The Z-score corresponding to a given level of confidence can be found from the Z-table or standard Normal Distribution. For a 90% confidence level, Z-score (Z) is 1.645.
03

Substitution and Calculation

Now, we substitute the known values into the formula: \(n = \left( \frac{1.645*25}{0.5} \right)^2 \). This simplifies to: \(n = 82.05^2 = 6726.2025 \). Because we cannot have a fraction of a sample, we will need to round up to the next whole number, providing a minimum sample size requirement.
04

Final Answer

Rounding up, the required minimum sample size (n) is 6727.

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Key Concepts

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

Confidence Interval
A confidence interval is a range of values that is used to estimate the true value of a population parameter. It is constructed around a sample statistic to capture the population parameter with a certain level of confidence, often expressed as a percentage like 90%, 95%, or 99%.

The width of this interval is influenced by the sample size, the level of confidence we desire, and the variability within the data, which is encapsulated by the standard deviation. The smaller the margin of error or the higher the confidence level we want, the larger the required sample size will be to ensure that our confidence interval accurately reflects the population mean.
Margin of Error
The margin of error represents the amount of error that one can tolerate in their estimate of a population parameter. It is the radius of the confidence interval and indicates the range above and below the sample statistic where the true population mean is expected to lie.

A smaller margin of error requires a larger sample size because it narrows the confidence interval, demanding more precision in the estimate. In statistical calculations, it is typically denoted by 'E' and directly affects how close the estimated interval is to the actual population value.
Normal Distribution
The normal distribution, often referred to as the bell curve, is a fundamental concept in statistics that describes how the values of a variable are distributed. It is characterized by its symmetrical, bell-shaped curve where most observations cluster around the central peak鈥攖he mean鈥攁nd probabilities for values taper off equally in both directions from the mean.

Many statistical methods, including the calculation of confidence intervals, rely on the assumption that data follow a normal distribution. This assumption allows statisticians to use the properties of the normal distribution to make inferences about population parameters based on sample data.
Z-score
A Z-score is a numerical measurement that describes a value's relationship to the mean of a group of values, measured in terms of standard deviations from the mean. If a Z-score is 0, it indicates that the data point's score is identical to the mean score.

In the context of confidence intervals, the Z-score is used to determine how many standard deviations an element is from the mean. When we state a 90% confidence level, we refer to the area under the normal distribution curve that captures 90% of the data, and the corresponding Z-score provides the critical value required for calculating the confidence interval.
Standard Deviation
The standard deviation (蟽) is a measure of how spread out numbers are in a set of data. It is a way to quantify the amount of variation or dispersion of a set of values. A low standard deviation means that the values tend to be close to the mean of the set, while a high standard deviation indicates that the values are spread out over a wider range.

In sample size determination for confidence intervals, the standard deviation reflects the inherent variability in the population. If the population variability is high, a larger sample will be required to achieve a certain margin of error for a given confidence level.

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Most popular questions from this chapter

Use a t-distribution. Assume the samples are random samples from distributions that are reasonably normally distributed, and that a t-statistic will be used for inference about the difference in sample means. State the degrees of freedom used. Find the area in a t-distribution above 2.1 if the samples have sizes \(n_{1}=12\) and \(n_{2}=12\).

Refer to a study on hormone replacement therapy. Until 2002 , hormone replacement therapy (HRT), taking hormones to replace those the body no longer makes after menopause, was commonly prescribed to post-menopausal women. However, in 2002 the results of a large clinical trial \(^{56}\) were published, causing most doctors to stop prescribing it and most women to stop using it, impacting the health of millions of women around the world. In the experiment, 8506 women were randomized to take HRT and 8102 were randomized to take a placebo. Table 6.16 shows the observed counts for several conditions over the five years of the study. (Note: The planned duration was 8.5 years. If Exercises 6.205 through 6.208 are done correctly, you will notice that several of the p-values are just below \(0.05 .\) The study was terminated as soon as HRT was shown to significantly increase risk (using a significance level of \(\alpha=0.05)\), because at that point it was unethical to continue forcing women to take HRT). Does HRT influence the chance of a woman getting invasive breast cancer? $$ \begin{array}{lcc} \hline \text { Condition } & \text { HRT Group } & \text { Placebo Group } \\ \hline \text { Cardiovascular Disease } & 164 & 122 \\ \text { Invasive Breast Cancer } & 166 & 124 \\ \text { Cancer (all) } & 502 & 458 \\ \text { Fractures } & 650 & 788 \\ \hline \end{array} $$

Use StatKey or other technology to generate a bootstrap distribution of sample means and find the standard error for that distribution. Compare the result to the standard error given by the Central Limit Theorem, using the sample standard deviation as an estimate of the population standard deviation. Mean body temperature, in \({ }^{\circ} \mathrm{F},\) using the data in BodyTemp50 with \(n=50, \bar{x}=98.26,\) and \(s=0.765\)

Is a Normal Distribution Appropriate? In Exercises 6.13 and \(6.14,\) indicate whether the Central Limit Theorem applies so that the sample proportions follow a normal distribution. In each case below, does the Central Limit Theorem apply? (a) \(n=80\) and \(p=0.1\) (b) \(n=25\) and \(p=0.8\) (c) \(n=50\) and \(p=0.4\) (d) \(n=200\) and \(p=0.7\)

Green Tea and Prostate Cancer A preliminary study suggests a benefit from green tea for those at risk of prostate cancer. \(^{55}\) The study involved 60 men with PIN lesions, some of which turn into prostate cancer. Half the men, randomly determined, were given \(600 \mathrm{mg}\) a day of a green tea extract while the other half were given a placebo. The study was double-blind, and the results after one year are shown in Table \(6.14 .\) Does the sample provide evidence that taking green tea extract reduces the risk of developing prostate cancer? $$ \begin{array}{lcc} \hline \text { Treatment } & \text { Cancer } & \text { No Cancer } \\ \hline \text { Green tea } & 1 & 29 \\ \text { Placebo } & 9 & 21 \end{array} $$
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