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Chapter 6: Problem 248

Use technology and the NutritionStudy dataset to find a \(95 \%\) confidence interval for the difference in number of grams of fiber \((\) Fiber \()\) eaten in a day between males and females. Interpret the answer in context. Is "No difference" between males and females a plausible option for the population difference in number of grams of fiber eaten?

Short Answer

Expert verified
The 95% confidence interval for the difference in fiber intake between males and females is calculated and interpreted in the context of the problem. The 'No difference' hypothesis is evaluated based on whether the confidence interval includes zero or not.

Step by step solution

01

Calculate Means

Use the NutritionStudy dataset to calculate the mean amount of fiber intake for both males and females, separately. Let's denote them as \( \mu_{males} \) and \( \mu_{females} \). Let's assume \( n_{males} \) and \( n_{females} \) as the number of samples in each group.
02

Calculate Standard Deviations

Calculate the standard deviations for the fiber intake for both groups, separately. Let's call them \( \sigma_{males} \) and \( \sigma_{females} \).
03

Calculate Standard Error

The standard error \( SE \) for the difference between these two means is calculated using this formula: \[ SE = \sqrt{ \frac{\sigma_{males}^2}{n_{males}} + \frac{\sigma_{females}^2}{n_{females}} } \]
04

Calculate Confidence Interval

To find the 95% confidence interval for the difference, calculate the margin of error using the standard error and the z-score for a 95% confidence level, which is 1.96. The confidence interval is given by: \[ CI = (\mu_{males} - \mu_{females}) \pm 1.96 \times SE \] This results in a range of values. This is our confidence interval.
05

Interpret the Results

The confidence interval represents the plausible values for the difference between the average fiber intake for males and females in the entire population. It is explained that we are 95% confident that the actual difference in mean fiber intake for all males and females lies within this range.
06

Evaluate 'No Difference' Hypothesis

`No difference` would mean that the difference in grams of Fiber eaten is zero. If the confidence interval includes zero, then it could be plausible to assume that there's no difference in the mean fiber intake between males and females in the population. If zero is not within this interval, we then reject the `No difference` hypothesis.

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Key Concepts

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

Statistical Hypothesis
In statistics, a hypothesis is an educated guess or claim about a population parameter, such as a mean or proportion. A hypothesis can be a statement like "There is no difference in mean fiber intake between males and females," which we call the null hypothesis. This particular hypothesis proposes that any observed difference is due to random chance rather than a genuine effect.

The opposite is the alternative hypothesis, which suggests that there is a true effect or difference. In this exercise, the alternative hypothesis would suggest a genuine difference in fiber intake between males and females. The purpose of hypothesis testing is to determine which hypothesis is more plausible based on the sample data collected. If our 95% confidence interval includes the value zero, the null hypothesis might be plausible, suggesting no real difference in the population means.
Mean Comparison
Mean comparison is a critical step in determining differences between two distinct groups. In this exercise, it involves comparing the average fiber intake between males and females.

To begin, calculate the means of both groups separately:
  • Determine the mean fiber intake for males: \( \mu_{males} \)
  • Determine the mean fiber intake for females: \( \mu_{females} \)
Next, calculate the difference between these two means. This difference gives an initial indication of whether there might be a variation in fiber intake between the two groups. However, a mere difference in sample means isn't enough to conclude a population-level difference. Instead, we use further statistical tests, creating a confidence interval, to determine if this difference is statistically significant.
Standard Deviation
Standard deviation provides a measure of how much variation exists in a dataset. In simpler terms, it tells us how spread out the numbers in a dataset are.

In the context of this exercise, we determine the standard deviation of fiber intake for both males and females separately:
  • Calculate the standard deviation for males: \( \sigma_{males} \)
  • Calculate the standard deviation for females: \( \sigma_{females} \)
Once known, these standard deviations help us calculate the standard error, which is crucial for forming the confidence interval. A larger standard deviation implies more variability in fiber intake, which can affect how confident we are about the mean differences between males and females. Understanding standard deviation is important for interpreting the confidence interval and making correct inferences regarding the hypothesis tested.

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Most popular questions from this chapter

In Exercises 6.153 to \(6.158,\) if random samples of the given sizes are drawn from populations with the given proportions: (a) Find the mean and standard error of the distribution of differences in sample proportions, \(\hat{p}_{A}-\hat{p}_{B}\) (b) If the sample sizes are large enough for the Central Limit Theorem to apply, draw a curve showing the shape of the sampling distribution. Include at least three values on the horizontal axis. Samples of size 50 from population \(A\) with proportion 0.70 and samples of size 75 from population \(B\) with proportion 0.60

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Drinking tea appears to offer a strong boost to the immune system. In a study introduced in Exercise 3.82 on page \(203,\) we see that production of interferon gamma, a molecule that fights bacteria, viruses, and tumors, appears to be enhanced in tea drinkers. In the study, eleven healthy non-tea- drinking individuals were asked to drink five or six cups of tea a day, while ten healthy nontea- and non-coffee-drinkers were asked to drink the same amount of coffee, which has caffeine but not the \(L\) -theanine that is in tea. The groups were randomly assigned. After two weeks, blood samples were exposed to an antigen and production of interferon gamma was measured. The results are shown in Table 6.23 and are available in ImmuneTea. The question of interest is whether the data provide evidence that production is enhanced in tea drinkers. (a) Is this an experiment or an observational study? (b) What are the null and alternative hypotheses? (c) Find a standardized test statistic and use the t-distribution to find the p-value and make a conclusion. (d) Always plot your data! Look at a graph of the data. Does it appear to satisfy a normality condition? (e) A randomization test might be a more appropriate test to use in this case. Construct a randomization distribution for this test and use it to find a p-value and make a conclusion. (f) What conclusion can we draw? $$ \begin{array}{lrrrrrr} \hline \text { Tea } & 5 & 11 & 13 & 18 & 20 & 47 \\ & 48 & 52 & 55 & 56 & 58 & \\ \hline \text { Coffee } & 0 & 0 & 3 & 11 & 15 & 16 \\ & 21 & 21 & 38 & 52 & & \\ \hline \end{array} $$

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