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Chapter 8: Problem 55

Estimating BMI The body mass index (BMI) of all American young women is believed to follow a Normal distribution with a standard deviation of about 7.5. How large a sample would be needed to estimate the mean BMl \(\mu\) in this population to within \(\pm 1\) with 99\(\%\) confidence? Show your work.

Short Answer

Expert verified
The required sample size is 374.

Step by step solution

01

Understanding the Problem

We need to determine the sample size required to estimate the mean BMI of American young women with a 99% confidence interval that is within 卤1 unit of BMI. The standard deviation is given as 7.5.
02

Identifying the Critical Value

For a 99% confidence level, we use a Z-distribution. The critical Z-value for 99% confidence is approximately 2.576. You can find this value in a standard Z-table.
03

Setting Up the Margin of Error Formula

The margin of error (E) formula is given by:\[ E = Z \cdot \frac{\sigma}{\sqrt{n}} \]where \( Z \) is the critical value, \( \sigma \) is the standard deviation (7.5), and \( n \) is the sample size. Here, \( E \) is given as 1.
04

Rearranging the Formula to Solve for Sample Size (n)

Rearrange the formula to solve for \( n \):\[ n = \left(\frac{Z \cdot \sigma}{E}\right)^2 \]Substitute the given values: \( Z = 2.576 \), \( \sigma = 7.5 \), and \( E = 1 \).
05

Calculating the Sample Size

Substitute the numbers into the equation:\[ n = \left(\frac{2.576 \cdot 7.5}{1}\right)^2 \]Calculate the result: \\[ = \left(19.32\right)^2 \]\[ = 373.7024 \]
06

Finalizing the Sample Size

Since sample size must be a whole number, we round up because a larger sample size will ensure the desired level of confidence. Thus, the required sample size is 374.

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Key Concepts

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

Confidence Interval
Confidence intervals provide a range of values that estimate an unknown population parameter, such as a mean or proportion.
Understanding them is key to evaluating how confident we are in our statistical estimates. When we talk about a confidence interval, we usually refer to a percentage that tells us how sure we are, for example, 99% in this instance.
  • A 99% confidence interval implies that if we were to take 100 different samples and compute a confidence interval for each, approximately 99 of those intervals would contain the true mean BMI.
  • This does not mean that there's a 99% chance that the calculated interval contains the population mean across all samples.
  • The remaining possibility, 1%, is the chance that our interval does not contain the true population mean.
In statistical practice, a higher confidence level requires a larger sample size if we want to keep the precision (margin of error) unchanged.
Normal Distribution
Normal distribution, also referred to as a Gaussian distribution, describes how the values of a variable are distributed. It is often depicted as a bell-shaped curve. In a normal distribution, most of the data points cluster around the mean.
As we move away from the mean, the frequency of the values exponentially decreases.
  • Such distributions are characterized by their symmetric shape, with the mean, median, and mode located at the center.
  • The spread of the distribution is determined by the standard deviation: a smaller standard deviation results in a taller and narrower curve, while a larger one leads to a wider and flatter curve.
  • In our problem, BMI follows this type of distribution, making our statistical calculations more manageable.
Many statistical methods, including estimating sample sizes and calculating confidences, rely on the properties of normal distribution.
Margin of Error
The margin of error provides an upper limit on how much we expect the sample estimate to vary from the true population parameter. It reflects the range within which the true parameter value lies.
  • A margin of error of 1 signifies that the sample mean is expected to be within 1 BMI unit of the actual population mean.
  • This is crucial because it informs us on the precision of our estimate. Smaller margins breed more precise estimates.
Calculating the margin of error involves this formula:\[ E = Z \cdot \frac{\sigma}{\sqrt{n}} \] Where:
  • E is the margin of error, supplied as 1.
  • \( Z \) is the Z-score associated with the chosen confidence level.
  • \( \sigma \) is the standard deviation of the population (here, 7.5).
  • \( n \) is the sample size.
With a given margin, we can determine the sample size necessary to maintain desired levels of precision.
Standard Deviation
Standard deviation is a measure that is used to quantify the amount of variation or dispersion of a set of values.
In simpler terms, it tells us how much our individual data points deviate from the average.
  • A smaller standard deviation means data points are more clustered around the mean, indicating less variability.
  • Inversely, a larger standard deviation denotes data points that are more spread out, showing higher variability.
The standard deviation directly influences the spread of the normal distribution and is pivotal in determining sample size. For example, with a standard deviation of 7.5 as in this problem, the data points (BMI measurements) are moderately spread around the mean. This aspect is integrated into calculations to see how large our sample needs to be to achieve an accurate estimate of the population mean while respecting our desired margin of error.

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Most popular questions from this chapter

How common is SAT coaching? A random sample of students who took the SAT college entrance examination twice found that 427 of the respondents had paid for coaching courses and that the remaining 2733 had not.\(^{14}\) Construct and interpret a 99\(\%\) confidence interval for the proportion of coaching among students who retake the SAT. Follow the four-step process.

Explaining confidence \(A 95 \%\) confidence interval for the mean body mass index (BMI) of young American women is \(26.8 \pm 0.6 .\) Discuss whether each of the following explanations is correct. (a) We are confident that 95% of all young women have BMI between 26.2 and 27.4. (b) We are 95% confident that future samples of young women will have mean BMI between 26.2 and 27.4. (c) Any value from 26.2 to 27.4 is believable as the true mean BMI of young American women. (d) In 95% of all possible samples, the population mean BMI will be between 26.2 and 27.4. (e) The mean BMI of young American women cannot be 28.

Blood pressure A medical study finds that \(\overline{x}=114.9\) and \(s_{x}=9.3\) for the seated systolic blood pressure of the 27 members of one treatment group. What is the standard error of the mean? Interpret this value in context.

For Exercises 27 to 30, check whether each of the conditions is met for calculating a confidence interval for the population proportion p. Whelks and mussels The small round holes you often see in sea shells were drilled by other sea creatures, who ate the former dwellers of the shells. Whelks often drill into mussels, but this behavior appears to be more or less common in different locations. Researchers collected whelk eggs from the coast of Oregon, raised the whelks in the laboratory, then put each whelk in a container with some delicious mussels. Only 9 of 98 whelks drilled into a mussel.11 The researchers want to estimate the proportion p of Oregon whelks that will spontaneously drill into mussels.

Multiple choice: Select the best answer for Exercises 21 to 24. You have measured the systolic blood pressure of an SRS of 25 company employees. A 95% confidence interval for the mean systolic blood pressure for the employees of this company is (122, 138). Which of the following statements gives a valid interpretation of this interval? (a) 95% of the sample of employees have a systolic blood pressure between 122 and 138. (b) 95% of the population of employees have a systolic blood pressure between 122 and 138. (c) If the procedure were repeated many times, 95% of the resulting confidence intervals would contain the population mean systolic blood pressure. (d) The probability that the population mean blood pressure is between 122 and 138 is 0.95. (e) If the procedure were repeated many times, 95% of the sample means would be between 122 and 138.
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