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Chapter 31: Problem 20

It is claimed that the two following sets of values were obtained (a) by randomly drawing from a normal distribution that is \(N(0,1)\) and then (b) randomly assigning each reading to one of two sets A and B: $$ \begin{array}{lrrrrrrr} \text { Set A: } & -0.314 & 0.603 & -0.551 & -0.537 & -0.160 & -1.635 & 0.719 \\\ & 0.610 & 0.482 & -1.757 & 0.058 & & & \\ \text { Set B: } & -0.691 & 1.515 & -1.642 & -1.736 & 1.224 & 1.423 & 1.165 \end{array} $$ Make tests, including \(t-\) and \(F\)-tests, to establish whether there is any evidence that either claims is, or both claims are, false.

Short Answer

Expert verified
Use t-test and F-test to compare means and variances. If p-values are high, no evidence to reject claims. If low, evidence to reject claims.

Step by step solution

01

Understand the Problem Statement

You are given two sets of values allegedly drawn from a normal distribution \( N(0,1) \), and it is claimed that these values were then randomly assigned to sets A and B. You need to test this claim using statistical tests like the t-test and F-test.
02

Calculate Basic Descriptive Statistics

Calculate the mean and standard deviation for both sets A and B. This will help in comparing the two sets and check if they come from the same distribution.
03

Perform t-test for Means

Use the t-test to compare the means of the two sets to determine if there is a statistically significant difference between them. The null hypothesis is that the means of the two sets are equal.
04

Perform F-test for Variances

Use the F-test to compare the variances of the two sets to determine if there is a statistically significant difference between them. The null hypothesis is that the variances of the two sets are equal.
05

Make a Conclusion

Based on the p-values from the t-test and F-test, conclude whether there is enough evidence to reject the claims that the values were drawn from a normal distribution \( N(0,1) \) and randomly assigned to sets A and B.

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Key Concepts

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

t-test
The t-test is a statistical hypothesis test used to compare the means of two groups. When you perform a t-test, you are essentially asking: *Are the average values of two groups significantly different from each other?* The null hypothesis in a t-test is that the means of the two groups are equal. If the p-value from the t-test is less than your chosen significance level (for example, 0.05), you can reject the null hypothesis. In the context of the exercise, after calculating the mean and standard deviation for both sets A and B, a t-test is performed to check if the average values differ significantly. This helps verify the claim that the sets were randomly assigned from a normal distribution. Remember, a low p-value indicates strong evidence against the null hypothesis, suggesting a significant difference in means.
F-test
The F-test is used to compare two variances to determine if they come from populations with equal variances. This is crucial when analyzing the variability within data sets. The null hypothesis for an F-test is that the variances of the two sets are equal. Similar to the t-test, if the p-value from the F-test is less than the significance level (like 0.05), you reject the null hypothesis. For example, in the exercise given, an F-test is used after calculating the variances for sets A and B. This helps determine if the assumption of equal variances holds true, which supports or contradicts the claim that data is randomly drawn from a normal distribution. A high p-value would fail to reject the null hypothesis, suggesting equal variances.
normal distribution
A normal distribution, also known as Gaussian distribution, is a bell-shaped graph. It is symmetric about the mean, indicating that data near the mean are more frequent in occurrence than data far from the mean. Properties include:
  • The mean, median, and mode are all equal.
  • The curve is symmetrical about the mean.
  • Approximately 68% of data falls within one standard deviation (\( \sigma \)) of the mean (\( \mu \)), 95% within two, and 99.7% within three.
The exercise involves testing whether sets A and B are drawn from a normal distribution N(0,1), meaning with mean (\( \mu \)) 0 and standard deviation (\( \sigma \)) 1. Identifying normal distribution characteristics can help validate the initial claim and appropriately apply the t-test and F-test.
descriptive statistics
Descriptive statistics provide a summary of the main characteristics of a data set. This includes measures like the mean, median, mode, and standard deviation. They help in understanding the basic properties of the data before applying more complex analyses. In the problem exercise, calculating the mean and standard deviation for sets A and B are initial steps. These values enable deeper statistical testing (e.g., t-test and F-test) to verify the claims about the data distribution. For instance, a significant difference in means or variances might question the claim that values are drawn from a normal distribution and randomly assigned to sets. Thus, descriptive statistics lay the groundwork for hypothesis testing.

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Most popular questions from this chapter

According to a particular theory, two dimensionless quantities \(X\) and \(Y\) have equal values. Nine measurements of \(X\) gave values of \(22,11,19,19,14,27,8\), 24 and 18, whilst seven measured values of \(Y\) were \(11,14,17,14,19,16\) and 14. Assuming that the measurements of both quantities are Gaussian distributed with a common variance, are they consistent with the theory? An alternative theory predicts that \(Y^{2}=\pi^{2} X ;\) are the data consistent with this proposal?

Measurements of a certain quantity gave the following values: \(296,316,307,278\), \(312,317,314,307,313,306,320,309\). Within what limits would you say there is a \(50 \%\) chance that the correct value lies?

A particle detector consisting of a shielded scintillator is being tested by placing it near a particle source whose intensity can be controlled by the use of absorbers. It might register counts even in the absence of particles from the source because of the cosmic ray background. The number of counts \(n\) registered in a fixed time interval as a function of the source strength \(s\) is given in as: \(\begin{array}{llrrrrrr}\text { source strength } s: & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ \text { counts } n: & 6 & 11 & 20 & 42 & 44 & 62 & 61\end{array}\) At any given source strength, the number of counts is expected to be Poisson distributed with mean $$ n=a+b s $$ where \(a\) and \(b\) are constants. Analyse the data for a fit to this relationship and obtain the best values for \(a\) and \(b\) together with their standard errors. (a) How well is the cosmic ray background determined? (b) What is the value of the correlation coefficient between \(a\) and \(b ?\) Is this consistent with what would happen if the cosmic ray background were imagined to be negligible? (c) Do the data fit the expected relationship well? Is there any evidence that the reported data 'are too good a fit'?

On a certain (testing) steeplechase course there are 12 fences to be jumped, and any horse that falls is not allowed to continue in the race. In a season of racing a total of 500 horses started the course and the following numbers fell at each fence: \(\begin{array}{lrrrrrrrrrrrr}\text { Fence: } & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\ \text { Falls: } & 62 & 75 & 49 & 29 & 33 & 25 & 30 & 17 & 19 & 11 & 15 & 12\end{array}\) Use this data to determine the overall probability of a horse's falling at a fence, and test the hypothesis that it is the same for all horses and fences as follows. (a) Draw up a table of the expected number of falls at each fence on the basis of the hypothesis. (b) Consider for each fence \(i\) the standardised variable $$ z_{i}=\frac{\text { estimated falls }-\text { actual falls }}{\text { standard deviation of estimated falls }} $$ and use it in an appropriate \(\chi^{2}\) test. (c) Show that the data indicates that the odds against all fences being equally testing are about 40 to 1 . Identify the fences that are significantly easier or harder than the average.

Prove that the sample mean is the best linear unbiased estimator of the population mean \(\mu\) as follows. (a) If the real numbers \(a_{1}, a_{2}, \ldots, a_{n}\) satisfy the constraint \(\sum_{i=1}^{n} a_{i}=C\), where \(C\) is a given constant, show that \(\sum_{i=1}^{n} a_{i}^{2}\) is minimised by \(a_{i}=C / n\) for all \(i\). (b) Consider the linear estimator \(\hat{\mu}=\sum_{i=1}^{n} a_{i} x_{i}\). Impose the conditions (i) that it is unbiased and (ii) that it is as efficient as possible.
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