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Chapter 6: Problem 94

A sample with \(n=18, \bar{x}=87.9,\) and \(s=10.6\)

Short Answer

Expert verified
Given the ambiguity of the problem, it is crucial to understand the specific statistical procedure or computation to be carried out. Therefore, the exercise should be read carefully and critically before proceeding to use the given \(n=18\), \(\bar{x}=87.9\), and \(s=10.6\) data.

Step by step solution

01

Given Information

The problem has not stated the required calculation(s) to be performed using the given statistical data, \(n=18\), \(\bar{x}=87.9\), and \(s=10.6\). These values represent the sample size, the average value of a data set, and the standard deviation or distribution from the mean, respectively.
02

Possible Applications of the Statistical Data

In the field of statistics, we can use this data to perform various tasks such as determining confidence intervals, testing hypotheses, or creating prediction intervals. In addition, the data can be used to compute the standard error of the mean, variance, or z-scores. However, it is important to note which statistical procedure to pursue depends on the problem or question at hand.

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Key Concepts

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

Sample Size
In descriptive statistics, the sample size is a critical concept. It refers to the number of observations or data points collected from a population to form a sample. In our exercise, the sample size is given as \(n = 18\). But why is this important?

The sample size is essential for a few reasons:
  • Accuracy of Results: A larger sample size typically leads to more accurate results. This is because a larger set reveals more about the population.
  • Variability Reduction: A small sample size may not accurately reflect the population鈥檚 characteristics, which can lead to higher variability in results.
  • Confidence Levels: The sample size affects the confidence level of any inferences made. With a larger sample, we can be more confident in the results.
Understanding sample size helps us make informed decisions in research. It's crucial not just to know the size but to understand its implications on the data and what conclusions we can draw from such data.
Standard Deviation
Standard deviation is a core concept in statistics that quantifies the amount of variation or dispersion in a set of data values. In the provided exercise, the standard deviation is \(s = 10.6\). But what does this tell us?
  • Measure of Spread: Standard deviation is a measure of how spread out the numbers in a data set are. A small standard deviation means the data points are close to the mean, while a large standard deviation indicates that the data points are more spread out.
  • Indicator of Consistency: If all data points are identical, the standard deviation is zero. More varied data will have a higher standard deviation, indicating less consistency.
  • Links to Other Statistics: Standard deviation is closely linked with variance, which is the square of the standard deviation. Both metrics are used to describe the distribution of the data.
In summary, knowing the standard deviation helps in understanding the reliability and consistency of the data, which can be especially important in making statistical inferences.
Confidence Intervals
Confidence intervals provide a range of values that likely contain a population parameter, such as the mean. But how do these intervals work?

In our example, though the exercise does not explicitly require it, we could use the given sample mean \(\bar{x} = 87.9\), the sample size \(n = 18\), and the standard deviation \(s = 10.6\) to construct a confidence interval. Here's what you need to know about them:
  • Estimate Precision: A confidence interval gives us a range of values for the sample mean where we can expect the population mean to fall.
  • Level of Confidence: This interval is associated with a confidence level (usually 95%), indicating how often this type of interval would capture the true population mean if we repeated the experiment many times.
  • Error Margin: A smaller interval indicates a more precise estimate of the population parameter. This precision depends on the sample size and standard deviation.
Understanding confidence intervals is key in statistical analysis as they allow researchers to make inferences about population parameters, maintaining a quantified degree of certainty.

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Most popular questions from this chapter

In the mid-1990s a Nabisco marketing campaign claimed that there were at least 1000 chips in every bag of Chips Ahoy! cookies. A group of Air Force cadets collected a sample of 42 bags of Chips Ahoy! cookies, bought from locations all across the country, to verify this claim. \({ }^{41}\) The cookies were dissolved in water and the number of chips (any piece of chocolate) in each bag were hand counted by the cadets. The average number of chips per bag was \(1261.6,\) with standard deviation 117.6 chips. (a) Why were the cookies bought from locations all over the country? (b) Test whether the average number of chips per bag is greater than 1000 . Show all details. (c) Does part (b) confirm Nabisco's claim that every bag has at least 1000 chips? Why or why not?

Exercise B.5 on page 305 introduces a study examining the effect of diet cola consumption on calcium levels in women. A sample of 16 healthy women aged 18 to 40 were randomly assigned to drink 24 ounces of either diet cola or water. Their urine was collected for three hours after ingestion of the beverage and calcium excretion (in mg) was measured. The summary statistics for diet cola are \(\bar{x}_{C}=56.0\) with \(s_{C}=4.93\) and \(n_{C}=8\) and the summary statistics for water are \(\bar{x}_{W}=49.1\) with \(s_{W}=3.64\) and \(n_{W}=8\). Figure 6.26 shows dotplots of the data values. Test whether there is evidence that diet cola leaches calcium out of the system, which would increase the amount of calcium in the urine for diet cola drinkers. In Exercise B.5, we used a randomization distribution to conduct this test. Use a t-distribution here, after first checking that the conditions are met and explaining your reasoning. The data are stored in ColaCalcium.

Rrefer 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 cardiovascular disease? $$ \begin{array}{lcc} \hline \text { Condition } & \text { HRT Group } & \text { Placebo Group } \\ \hline \text { Cardiovascular Disease } & 164 & 122 \\ \text { Invasive Breast Cancer } & 166 & 124 \\ \text { Cancer (all) } & 502 & 458 \\ \text { Fractures } & 650 & 788 \\ \hline \end{array} $$

Use 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 endpoints of the t-distribution with \(5 \%\) beyond them in each tail if the samples have sizes \(n_{1}=8\) and \(n_{2}=10\)

A sample with \(n=12, \bar{x}=7.6,\) and \(s=1.6\)
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