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Chapter 27: 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 the two sets \(\mathrm{A}\) and \(\mathrm{B}\). \(\begin{array}{lrrrrrrr}\text { Set A } & -0.314 & 0.603 & -0.551 & -0.537 & -0.160 & -1.635 & 0.719 \\ & 0.610 & 0.482 & -1.757 & 0.058 & & & \\ \text { Set B } & -0.691 & 1.515 & -1.642 & -1.736 & 1.224 & 1.423 & 1.165\end{array}\) 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
Perform a t-test and F-test to compare means and variances. If no significant differences are found, both claims may be true.

Step by step solution

01

- Calculate Means of Sets A and B

First, calculate the means of sets A and B. The mean is the sum of all values divided by the number of values.
02

- Calculate Variances of Sets A and B

Calculate the variances of sets A and B. Variance is the average of the squared differences from the mean.
03

- Perform the Two-Sample t-Test

Use the sample means, sample standard deviations, and sizes of Sets A and B to calculate the t-statistic. Compare the calculated t-value with the critical value from the t-distribution table at a chosen significance level.
04

- Perform the F-Test for Equal Variances

Calculate the F-statistic to compare variances of Sets A and B. The F-statistic is the ratio of the larger sample variance to the smaller sample variance. Compare the calculated F-value with the critical value from the F-distribution table.
05

- Interpret Results

Analyze the results of the t-test and F-test. If the t-test shows no significant difference in means and the F-test indicates equal variances, both claims may hold true.

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Key Concepts

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

normal distribution
A normal distribution is a bell-shaped curve that is symmetrical around the mean. This distribution is often used in statistics because many real-world phenomena tend to follow this pattern.
Key properties of a normal distribution include:
  • Mean, median, and mode are all equal
  • Symmetrical shape
  • Approach zero asymptotically in both directions
In the exercise, it's claimed the values were drawn from a normal distribution with mean \(\text{N}(0,1)\). This means the values should center around 0 with a standard deviation of 1.
t-test
The t-test is used to determine if there are significant differences between the means of two datasets. In this context, we perform a two-sample t-test.
Here's how it works:
  • Calculate the means of sets A and B
  • Compute the variances and standard deviations of both sets
  • Determine the number of observations in each set
  • Use the formula for the t-statistic: \(\text{t} = \frac{\bar{X}_1 - \bar{X}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}\)
The calculated t-value is then compared with the critical value from the t-distribution table. If the t-value falls within the critical range, we cannot reject the null hypothesis, indicating no significant difference between the means.
F-test
An F-test compares the variances of two datasets to see if they are significantly different. This is important for determining the consistency in data dispersion across samples.
Steps to perform an F-test:
  • Calculate the variances of sets A and B
  • Determine the ratio of the larger variance to the smaller variance. This ratio is the F-statistic: \(\text{F} = \frac{s_1^2}{s_2^2}\)
The F-statistic is compared with the critical value from the F-distribution table at a chosen significance level. If the ratio is within the critical value, we assume the variances are equal.
sample mean
The sample mean is the average of all values in a dataset. It's a measure of central tendency that provides a quick summary of the data.
To calculate the sample mean:
  • Sum all values in the dataset
  • Divide by the number of values
In the exercise, you calculate the means for sets A and B to analyze their central tendencies and to use them in further statistical tests like the t-test.
variance
Variance measures how much the values in a dataset differ from the mean. It provides an idea of the data's spread.
Steps to calculate variance:
  • Find the mean of the dataset
  • Subtract the mean from each value to find the deviation
  • Square each deviation
  • Calculate the average of these squared deviations. This is the variance: \(\text{Variance} = \frac{\sum (X - \bar{X})^2}{n-1}\)
Variance is crucial for performing an F-test and understanding the dispersion of data in sets A and B.

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Most popular questions from this chapter

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.

The function \(y(x)\) is known to be a quadratic function of \(x\). The following table gives the measured values and uncorrelated standard errors of \(y\) measured at various values of \(x\) (in which there is negligible error): $$ \begin{array}{lccccc} x & 1 & 2 & 3 & 4 & 5 \\ y(x) & 3.5 \pm 0.5 & 2.0 \pm 0.5 & 3.0 \pm 0.5 & 6.5 \pm 1.0 & 10.5 \pm 1.0 \end{array} $$ Construct the response matrix \(R\) using as basis functions \(1, x, x^{2} .\) Calculate the matrix \(\mathrm{R}^{\mathrm{T}} \mathrm{N}^{-1} \mathrm{R}\) and show that its inverse, the covariance matrix \(\mathrm{V}\), has the form $$ \mathrm{V}=\frac{1}{9184}\left(\begin{array}{ccc} 12592 & -9708 & 1580 \\ -9708 & 8413 & -1461 \\ 1580 & -1461 & 269 \end{array}\right). $$ Use this matrix to find the best values, and their uncertainties, for the coefficients of the quadratic form for \(y(x)\).

An experiment consists of a large, but unknown, number \(n(\gg 1)\) of trials in each of which the probability of success \(p\) is the same, but also unkown. In the \(i\) th trial, \(i=1,2, \ldots, N\), the total number of successes is \(x_{i}(\gg 1)\). Determine the log-likelihood function. Using Stirling's approximation to \(\ln (n-x)\), show that $$ \frac{d \ln (n-x)}{d n} \approx \frac{1}{2(n-x)}+\ln (n-x) $$ and hence evaluate \(\partial\left({ }^{n} C_{x}\right) / \partial n\) By finding the (coupled) equations determining the ML estimators \(\hat{p}\) and \(\hat{n}\), show that, to order \(N^{-1}\), they must satisfy the simultaneous 'arithmetic' and 'geometric' mean constraints $$ \hat{n} \hat{p}=\frac{1}{N} \sum_{i=1}^{N} x_{i} \quad \text { and } \quad(1-\hat{p})^{N}=\prod_{i=1}^{N}\left(1-\frac{x_{i}}{\hat{n}}\right). $$

A particle detector consisting of a shielded scintillator is being tested by placing it near a particle source of controlled intensity (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 the following table: $$ \begin{array}{llrrrrrr} \text { Source strength: } s & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ \text { Counts } n: & 6 & 11 & 20 & 42 & 44 & 62 & 61 \end{array} $$ 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) Does the data fit the expected relationship well? Is there any evidence that the reported data 'is too good a fit'?

Two physical quantities \(x\) and \(y\) are connected by the equation $$ y^{1 / 2}=\frac{x}{a x^{1 / 2}+b} $$ and measured pairs of values for \(x\) and \(y\) are as follows: \(\begin{array}{rrrrr}x: & 10 & 12 & 16 & 20 \\ y: & 409 & 196 & 114 & 94\end{array}\) Determine the best values for \(a\) and \(b\) by graphical means and (either by hand or by using a built-in calculator routine) by a least squares fit to an appropriate straight line,
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