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Chapter 10: Problem 94

Show that the standard error of \(p_{1}^{\prime}-p_{2}^{\prime},\) which is \(\sqrt{\left(\frac{p_{1} q_{1}}{n_{1}}\right)+\left(\frac{p_{2} q_{2}}{n_{2}}\right)},\) reduces to \(\sqrt{p q\left[\left(\frac{1}{n_{1}}\right)+\left(\frac{1}{n_{2}}\right)\right]}\) when \(p_{1}=p_{2}=p\)

Short Answer

Expert verified
The standard error \(\sqrt{\left(\frac{p_{1} q_{1}}{n_{1}}\right)+\left(\frac{p_{2}q_{2}}{n_{2}}\right)}\) simplifies to \(\sqrt{pq\left[\left(\frac{1}{n_{1}}\right)+\left(\frac{1}{n_{2}}\right)\right]}\) when \(p_{1}=p_{2}=p\)

Step by step solution

01

Write down the given equations

We have the standard error formula as: \[ SE = \sqrt{\left(\frac{p_{1} q_{1}}{n_{1}}\right)+\left(\frac{p_{2} q_{2}}{n_{2}}\right)} \]And we are given that \( p_{1} = p_{2} = p \) and thus \( q_{1} = q_{2} = q \) where \( q = 1 - p \)
02

Substitute the given equations into the standard error formula

Replace the \( p_{1} \), \( p_{2} \), \( q_{1} \), and \( q_{2} \) with \( p \) and \( q \) in the standard error formula. This gives us: \[ SE = \sqrt{\left(\frac{p q}{n_{1}}\right)+\left(\frac{p q}{n_{2}}\right)} \]
03

Factor out the common terms

Note that \( p q \) is a common term in the two expressions within the square root. Factoring it out, we get: \[ SE = \sqrt{pq\left[\left(\frac{1}{n_{1}}\right)+\left(\frac{1}{n_{2}}\right)\right]} \]

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Key Concepts

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

Understanding Hypothesis Testing
Hypothesis testing is a statistical method used to make decisions about the properties of a population based on sample data. It's like a detective at work, using clues to draw conclusions. In hypothesis testing, you start with a null hypothesis, which is a statement assuming no effect or no difference, and an alternative hypothesis, which is what you want to prove. For example, the null hypothesis might state that there is no difference between two population proportions, while the alternative suggests a significant difference exists.
  • First, you decide on a significance level, usually 0.05, which is the probability of rejecting the null hypothesis when it is true.
  • Then, data is collected and a test statistic calculated, which measures the agreement between the sample data and the null hypothesis.
  • Finally, you compare this statistic to a critical value to determine whether to reject or fail to reject the null hypothesis.
Hypothesis testing helps you make informed decisions by quantifying uncertainty, making it a cornerstone in statistics.
Exploring Sampling Distribution
Sampling distribution describes how a sample statistic (like the sample mean or sample proportion) behaves when it is drawn repeatedly from the same population. Imagine drawing many samples from a bowl of jellybeans and recording the proportion of red jellybeans each time; this collection of proportions forms a sampling distribution.
  • The shape of a sampling distribution depends on the sample size and the true population proportion. As the sample size increases, the sampling distribution becomes more normal according to the Central Limit Theorem.
  • Its mean is equal to the population parameter being estimated, and its dispersion, measured by standard error, shows how much variability exists between samples.
Understanding the sampling distribution helps in predicting how much sampling fluctuation to expect and is crucial in constructing confidence intervals and conducting hypothesis tests.
Delving into Proportions
Proportions in statistics are like pieces of a puzzle, representing the fraction of a sample or population with a particular characteristic. For example, if 60 out of 100 surveyed people like chocolate, the proportion is 0.6. When comparing two proportions, such as the proportion of chocolate lovers in two different cities, understanding these proportions becomes key in hypothesis testing.
  • The difference between two sample proportions is often analyzed to see if it reflects a true difference in the populations.
  • Standard error plays a role here, as it quantifies how much the sample proportions would differ due to chance alone.
  • The formula for standard error when comparing two proportions, as seen in the exercise, can be simplified under the assumption that the population proportions are equal.
Grasping proportions and their behavior is fundamental in analyzing categorical data and drawing meaningful conclusions.

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Most popular questions from this chapter

A Harris Interactive poll found that \(50 \%\) of Democrats follow professional football while \(59 \%\) of Republicans follow the sport. If the poll results were based on samples of 875 Democrats and 749 Republicans, determine, at the 0.05 level of significance, if the viewpoint of more Republicans following professional football is substantiated.

Determine the test criteria that would be used with the classical approach to test the following hypotheses when \(t\) is used as the test statistic. a. \(\quad H_{o}: \mu_{d}=0\) and \(H_{a}: \mu_{d}>0,\) with \(n=15\) and \(\alpha=0.05\) b. \(\quad H_{o}: \mu_{d}=0\) and \(H_{a}: \mu_{d} \neq 0,\) with \(n=25\) and \(\alpha=0.05\) c. \(\quad H_{o}: \mu_{d}=0\) and \(H_{a}: \mu_{d}<0,\) with \(n=12\) and \(\alpha=0.10\) d. \(\quad H_{o}: \mu_{d}=0.75\) and \(H_{a}: \mu_{d}>0.75,\) with \(n=18\) and \(\alpha=0.01\)

Penfield and Perinton are two adjacent eastside suburbs of Rochester, New York. They have both always been considered on equal ground with respect to quality of life, housing, and education. Although many new housing developments are going up in Penfield, Perinton offers the added bonus of cheaper energy rates. It is thought that Perinton is able to have higher asking prices because of this bonus. To test this theory, random samples of real estate transactions were taken in each suburb during the week of October 17 \(2009 .\) Do the data support the theory for this time frame? Use \(\alpha=0.10\) $$\begin{array}{lc} \text { Pentield } & \text { Perinton } \\ \hline 195.700 & 154,900 \\\137,500 & 429,000 \\\117,000 & 272,000 \\\115,000 & 160,609 \\\176,000 & 265,000 \\\149,013 & 144,000 \\\130,000 & 152,000 \\ 266,490 & 390,000 \\\152,000 & 130,300 \\\262,765 & 149,900 \\\\\hline\end{array}$$

State the null hypothesis, \(H_{o}\), and the alternative hypothesis, \(H_{a},\) that would be used to test these claims: a. There is no difference between the proportions of men and women who will vote for the incumbent in next month's election. b. The percentage of boys who cut classes is greater than the percentage of girls who cut classes. c. The percentage of college students who drive old cars is higher than the percentage of noncollege people of the same age who drive old cars.

State the null and alternative hypotheses that would be used to test the following claims: a. There is a difference between the mean age of employees at two different large companies. b. The mean of population 1 is greater than the mean of population 2 c. The mean yield of sunflower seeds per county in North Dakota is less than the mean yield per county in South Dakota. d. There is no difference in the mean number of hours spent studying per week between male and female college students.
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