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Chapter 11: Problem 8

Explain the difference between the observed and expected frequencies for a goodness-of-fit test.

Short Answer

Expert verified
The observed frequencies are actual data collected from an experiment or a study. These are the real-counted occurrences. On the other hand, the expected frequencies are the theoretically predicted or calculated values based on an assumption or hypothesis. In a goodness-of-fit test, the observed and expected frequencies are compared to see if our observed outcomes deviate from what we were expecting, which could lead to the rejection of the null hypothesis.

Step by step solution

01

Understanding Observed Frequency

The observed frequency is the actual data that has been collected from an experiment or study. It represents the counted occurrence of different outcomes within a given dataset.
02

Understanding Expected Frequency

The expected frequency represents the frequency that we would expect to observe in each category if the proposed theory about the population is correct. It's typically based on theoretical probability distributions, which are mathematical models that describe how data behaves.
03

Understanding The Goodness-Of-Fit Test

The goodness-of-fit test is a statistical hypothesis test that is used to compare the observed distribution of data with an expected distribution of data, specified by a null hypothesis.
04

Identifying The Difference

The main difference between the observed and expected frequencies for a 'goodness-of-fit' test lies in their source of origin. The observed frequencies are derived directly from the data observed from the sample, while expected frequencies are based on a theoretical assumption under the null hypothesis.

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Key Concepts

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

Observed Frequency
Observed frequency is all about the actual data you gather from experiments or surveys. Imagine you鈥檙e counting the number of red, blue, and green marbles you pull from a bag. The tally you make as you pull each marble represents the observed frequency.
  • Observed frequency is the real count of occurrences within a dataset.
  • Helps in understanding what actually happens in your experiment.
  • Collected data can vary from one experiment to another.
It is crucial because it serves as the evidence you have to test your predictions or expectations. While it can differ in every trial based on a variety of factors, having accurate and precise data gathering techniques ensures that the observed frequencies are reliable for analysis.
Expected Frequency
Expected frequency brings theory into the picture. It predicts what you should observe if your assumptions about the data are correct. Picture it like a well-informed guess based on mathematical models such as probability distributions.
  • Expected frequency relies on theoretical models, often assuming a certain distribution pattern.
  • It helps to provide a benchmark to compare against what you observe.
  • Calculations are usually derived from probability theories like the normal distribution or binomial distribution.
These frequencies are not gathered from data - instead, they are calculated from models that attempt to describe the underlying process that generates the data. This makes them essential in assessing how realistic the patterns you observe in data are.
Statistical Hypothesis Test
Statistical hypothesis testing, including the goodness-of-fit test, allows us to put our assumptions to the test.
The main idea is to compare two things: what we observe (observed frequency) versus what we expect (expected frequency) under the null hypothesis. This is essential in determining if the pattern of data follows the expected theoretical model.
  • The goodness-of-fit test helps to decide if there is a significant difference between observed and expected frequencies.
  • Hypotheses involved usually include a null hypothesis that the data follows a specified distribution pattern.
  • Rejections of the null hypothesis imply a poor fit between theoretical model and observed data.
Using test statistics, we evaluate whether any observed discrepancies are due to random chance or represent significant deviations, guiding our decisions and interpretations of data.
Theoretical Probability Distribution
Theoretical probability distributions play a pivotal role in many statistical analyses, including determining expected frequencies. They are mathematical functions that describe the likelihood of various outcomes in a random experiment.
  • Distributions like normal, binomial, or Poisson are commonly used models.
  • They help predict outcomes and provide a framework for the expected frequency.
  • Understanding these distributions aids in assessing how well the observed frequencies align with what theory suggests.
These distributions are essential in making inferences about populations from samples, providing a crucial link between theory and practice in statistical hypothesis tests. By comparing the observed data to these theoretical models, we assess the validity of assumptions and theories.

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Most popular questions from this chapter

A company manufactures ball bearings that are supplied to other companies. The machine that is used to manufacture these ball bearings produces them with a variance of diameters of \(.025\) square millimeter or less. The quality control officer takes a sample of such ball bearings quite often and checks, using confidence intervals and tests of hypotheses, whether or not the variance of these bearings is within \(.025\) square millimeter. If it is not, the machine is stopped and adjusted. A recently taken random sample of 23 ball bearings gave a variance of the diameters equal to \(.034\) square millimeter. a. Using a \(5 \%\) significance level, can you conclude that the machine needs an adjustment? Assume that the diameters of all ball bearings have a normal distribution. b. Construct a \(95 \%\) confidence interval for the population variance.

In a Harris Poll conducted October \(15-20,2014\), American adults were asked "to think ahead 2 to 5 years and assess if they feel solar energy will contribute to meeting our energy needs." Of the respondents, \(31 \%\) said solar energy will make a major contribution to meeting our energy needs within the next 2 to 5 years, \(53 \%\) felt it will make a minor contribution, and \(16 \%\) expected that it will make hardly any contribution at all (www.harrisinteractive.com). Assume that these results are true for the 2014 population of adults. Recently a random sample of 2000 American adults was selected and these adults were asked the same question. The results of the poll are presented in the following table. $$ \begin{array}{l|ccc} \hline \text { Response } & \begin{array}{c} \text { Major } \\ \text { Contribution } \end{array} & \begin{array}{c} \text { Minor } \\ \text { Contribution } \end{array} & \begin{array}{c} \text { Hardly Any } \\ \text { Contribution } \end{array} \\ \hline \text { Frequency } & 820 & 920 & 260 \\ \hline \end{array} $$ Test at a \(2.5 \%\) significance level whether the current distribution of opinions to the said question is significantly different from that for the 2014 opinions.

The following table lists the frequency distribution for 60 rolls of a die. $$ \begin{array}{l|cccccc} \hline \text { Outcome } & 1 \text { -spot } & 2 \text { -spot } & \text { 3-spot } & 4 \text { -spot } & \text { 5-spot } & \text { 6-spot } \\ \hline \text { Frequency } & 7 & 12 & 8 & 15 & 11 & 7 \\ \hline \end{array} $$ Test at a \(5 \%\) significance level whether the null hypothesis that the given die is fair is true.

Chance Corporation produces beauty products. Two years ago the quality control department at the company conducted a survey of users of one of the company's products. The survey revealed that \(53 \%\) of the users said the product was excellent, \(31 \%\) said it was satisfactory, \(7 \%\) said it was unsatisfactory, and \(9 \%\) had no opinion. Assume that these percentages were true for the population of all users of this product at that time. After this survey was conducted, the company redesigned this product. A recent survey of 800 users of the redesigned product conducted by the quality control department at the company showed that 495 of the users think the product is excellent, 255 think it is satisfactory, 35 think it is unsatisfactory, and 15 have no opinion. Is the percentage distribution of the opinions of users of the redesigned product different from the percentage distribution of users of this product before it was redesigned? Use \(\alpha=.025\).

Home Mail Corporation sells products by mail. The company's management wants to find out if the number of orders received at the company's office on each of the 5 days of the week is the same. The company took a sample of 400 orders received during a 4 -week period. The following table lists the frequency distribution for these orders by the day of the week. $$ \begin{array}{l|ccccc} \hline \text { Day of the week } & \text { Mon } & \text { Tue } & \text { Wed } & \text { Thu } & \text { Fri } \\ \hline \begin{array}{l} \text { Number of orders } \\ \text { received } \end{array} & 92 & 71 & 65 & 83 & 89 \\ \hline \end{array} $$ Test at a \(5 \%\) significance level whether the null hypothesis that the orders are evenly distributed over all days of the week is true.
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