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Chapter 6: Problem 182

For each scenario, use the formula to find the standard error of the distribution of differences in sample means, \(\bar{x}_{1}-\bar{x}_{2}\) Samples of size 50 from Population 1 with mean 3.2 and standard deviation 1.7 and samples of size 50 from Population 2 with mean 2.8 and standard deviation 1.3

Short Answer

Expert verified
The standard error of the distribution of differences in sample means is 0.3035.

Step by step solution

01

Identify the Given Values

First, identify the given values. Here, for Population 1, the sample size (n1) is 50, the mean (碌1) is 3.2 and the standard deviation (s1) is 1.7. For Population 2, the sample size (n2) is 50, the mean (碌2) is 2.8 and the standard deviation (s2) is 1.3.
02

Apply the Formula

Next, apply the Standard Error of the difference between two means formula: SE = \(\sqrt{{\frac{{s1^2}}{{n1}} + \frac{{s2^2}}{{n2}}}}\). Substituting the given values into the formula, we get SE = \(\sqrt{{\frac{{(1.7)^2}}{{50}} + \frac{{(1.3)^2}}{{50}}}}\).
03

Calculate the Standard Error

Now, calculate the value inside the square root: \((1.7)^2 / 50 = 0.0578\) and \((1.3)^2 / 50 = 0.0343\). Adding these two values together we get \(0.0578 + 0.0343 = 0.0921\). Taking the square root of 0.0921, we find SE = 0.3035.

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Key Concepts

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

Distribution of Differences
Understanding the distribution of differences in sample means is crucial for comparing two populations. When you have two sets of data, like in our exercise, you often want to know if there is a significant difference between them. This is where the distribution of differences comes into play.

In our example, we take samples from two different populations. By calculating the standard error of these differences, we evaluate how much we expect the sample means to vary from each other due to random sampling effects.
  • It's like asking: Are the differences between the means due to actual differences in the populations, or just random chance?
  • The larger the standard error, the more variability, and hence, less confidence in distinguishing the difference between population means.
Think of this distribution as a way to visualize and quantify how much overlap there is between the two sample means. If they overlap significantly, it's harder to state there's a genuine difference between the populations.
Sample Means
Sample means are an estimate of the true population mean. In the given problem, sample means are taken from two populations: Population 1 and Population 2, each with 50 samples.

The sample mean is a point estimate of what the average of the entire population might be. Since it's impractical to measure the whole population, researchers use sample means.
  • For Population 1, the mean is 3.2.
  • For Population 2, the mean is 2.8.
By using these sample means, we can reason about the populations as a whole. Since no population is identical, sample means can tell us, with a level of confidence, how representative the sample is of the population. This is what makes sample means so powerful in statistical analysis.
Population Comparison
When comparing populations, like in our exercise, we're looking to understand if there is a statistically significant difference between them. The standard error helps us in that assessment.

The true power lies in comparing whether the differences in sample means reflect true population differences.
  • Smaller standard errors imply that the observed differences in means are more likely due to actual differences between populations, not random chance.
  • Larger standard errors suggest the opposite, implying any observed differences could easily be due to sampling variability.
In our specific problem, the standard error value 0.3035 helps us gauge this difference. A smaller standard error might suggest that the mean of Population 1 genuinely differs from that of Population 2. By determining these comparisons, researchers can make informed decisions, validate hypotheses, and answer compelling questions about population characteristics.

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Most popular questions from this chapter

Using Data 5.1 on page \(375,\) we find a significant difference in the proportion of fruit flies surviving after 13 days between those eating organic potatoes and those eating conventional (not organic) potatoes. ask you to conduct a hypothesis test using additional data from this study. \(^{40}\) In every case, we are testing $$\begin{array}{ll}H_{0}: & p_{o}=p_{c} \\\H_{a}: & p_{o}>p_{c}\end{array}$$ where \(p_{o}\) and \(p_{c}\) represent the proportion of fruit flies alive at the end of the given time frame of those eating organic food and those eating conventional food, respectively. Also, in every case, we have \(n_{1}=n_{2}=500 .\) Show all remaining details in the test, using a \(5 \%\) significance level. Effect of Organic Potatoes after 20 Days After 20 days, 250 of the 500 fruit flies eating organic potatoes are still alive, while 130 of the 500 eating conventional potatoes are still alive.

In Exercise \(6.107,\) we see that plastic microparticles are contaminating the world's shorelines and that much of the pollution appears to come from fibers from washing polyester clothes. The same study referenced in Exercise 6.107 also took samples from ocean beaches. Five samples were taken from each of 18 different shorelines worldwide, for a total of 90 samples of size \(250 \mathrm{~mL}\). The mean number of plastic microparticles found per \(250 \mathrm{~mL}\) of sediment was 18.3 with a standard deviation of 8.2 . (a) Find and interpret a \(99 \%\) confidence interval for the mean number of polyester microfibers per \(250 \mathrm{~mL}\) of beach sediment. (b) What is the margin of error? (c) If we want a margin of error of only 卤1 with \(99 \%\) confidence, what sample size is needed?

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 proportion in a t-distribution less than -1.4 if the samples have sizes \(n_{1}=30\) and \(n_{2}=40\)

Who Eats More Fiber: Males or Females? 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 mean number of grams of fiber eaten?

Find a \(95 \%\) confidence interval for the mean two ways: using StatKey or other technology and percentiles from a bootstrap distribution, and using the t-distribution and the formula for standard error. Compare the results. Mean price of a used Mustang car online, in \$1000s, using data in MustangPrice with \(\bar{x}=15.98\), \(s=11.11,\) and \(n=25\)
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