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Chapter 9: Problem 86

Make the following hypothesis tests about \(p\). a. \(H_{0}: p=.57, \quad H_{1}: p \neq .57, \quad n=800, \quad \hat{p}=.50, \quad \alpha=.05\) b. \(H_{0}=p=.26, \quad H_{1}: p<.26, \quad n=400, \quad \hat{p}=.23, \quad \alpha=.01\) c. \(H_{0}=p=.84, \quad H_{1}: p>.84, \quad n=250, \quad \hat{p}=.85, \quad \alpha=.025\)

Short Answer

Expert verified
a. Reject the null hypothesis; there is enough evidence that \( p \neq .57 \).\nb. Do not reject the null hypothesis; there isn't enough evidence that \( p < .26 \). \nc. Do not reject the null hypothesis; there isn't enough evidence that \( p > .84 \).

Step by step solution

01

Identify the type of test for each case

a. Since the alternative hypothesis is \( p \neq .57 \), it's a two-tailed test.\nb. Since the alternative hypothesis is \( p < .26 \), it's a left-tailed test.\nc. Since the alternative hypothesis is \( p > .84 \), it's a right-tailed test.
02

Compute the standard deviation and test statistic for each case

First, calculate the standard deviation: \( \sqrt{ \frac{p_{0}(1-p_{0})}{n} } \).\nThen, the test statistic: \( z = \frac{\hat{p} - p_{0}}{standard \ deviation} \).\na. Standard Deviation: \( \sqrt{ \frac{.57(1-.57)}{800} } = 0.0196\); Test Statistic: \( z = \frac{.50-.57}{0.0196} = -3.57 \)\nb. Standard Deviation: \( \sqrt{ \frac{.26(1-.26)}{400} } = 0.0221\); Test Statistic: \( z = \frac{.23-.26}{0.0221} = -1.36 \)\nc. Standard Deviation: \( \sqrt{ \frac{.84(1-.84)}{250} } = 0.0221\); Test Statistic: \( z = \frac{.85-.84}{0.0221} = 0.45 \)
03

Determine the critical value(s) and make a decision for each case

For a two-tailed test with \( \alpha = 0.05 \), the critical values are \(-1.96, +1.96\).\nFor a left-tailed test with \( \alpha = 0.01 \), critical value is \(-2.33\).\nFor a right-tailed test with \( \alpha = 0.025 \), critical value is \(+1.96\).\na. Test statistic \(-3.57 < -1.96\), reject \(H_{0}\). \nb. Test statistic \(-1.36 > -2.33\), fail to reject \(H_{0}\). \nc. Test statistic \(0.45 < 1.96\), fail to reject \(H_{0}\).

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Key Concepts

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

Two-tailed Test
In a two-tailed test, we are interested in detecting any "significant" difference, no matter the direction. Here, the null hypothesis asserts that the population parameter equals a specific value. Conversely, the alternative hypothesis states that the parameter is different from that value, either greater or lesser.
This test becomes relevant when deviations could occur in both directions at the point of interest.
  • Example: You want to test if the proportion of voters supporting a candidate is precisely 57%. If it's actually either more or less, it's significant if you're using a two-tailed test.
  • Common descriptive setting: Difference Hypothesized=True, so it can be both playing field.
Two-tailed tests generally involve setting up the hypothesis as: \[H_{0}: p = p_{0} \] \[H_{1}: p eq p_{0}\]
So, both tails of the distribution are taken into account when determining the critical values.
Left-tailed Test
In a left-tailed test, the focus is on whether a parameter is significantly less than the hypothesized value.
This type of test is applicable when only negative deviations from the hypothesized parameter matter for the hypothesis test.
  • Example: You might use a left-tailed test if you suspect the proportion of students passing a course is less than 26% as posited.
  • Common application: Seeking reduction from status quo or target number/value.
The hypotheses are structured as follows:\[H_{0}: p = p_{0}\] \[H_{1}: p < p_{0}\]
In such cases, the critical value is typically located on the left or "lower" tail of the distribution, explaining the name of this test.
Right-tailed Test
Conversely, a right-tailed test investigates the possibility that a parameter is greater than the hypothesized value.
It's useful when only positive deviations are significant.
  • Example: Suppose you expect more than 84% of a species' population to meet age maturity in captivity; a right-tailed test will suit this analysis.
  • Common context: Estimating improved outcomes or gains.
For right-tailed tests, the hypotheses are framed like this: \[H_{0}: p = p_{0}\] \[H_{1}: p > p_{0}\]
This test places emphasis on the right or "upper" tail of the probability distribution to determine significance.
Test Statistic
The test statistic is central whenever a hypothesis is being tested.
It measures how far the sampled data is from the null hypothesis.Usually represented by a "z-score" or "t-score," it standardizes the difference between the sample statistic and the hypothesized parameter.
  • Calculation formula: \[ z = \frac{\hat{p} - p_{0}}{\text{Standard deviation}} \]
  • Purpose: Evaluates extremity of sample results assuming null hypothesis is true.
Test statistics allow us to compare against critical values, determining whether we should reject or fail to reject the null hypothesis.
Critical Value
A critical value is a point on the test distribution beyond which we reject the null hypothesis.
These values are determined based on the alpha level, which is the probability threshold for significance (commonly known as significance level).
  • Standard critical values:
    • Two-tailed at \(\alpha = 0.05\) yields critical values of \(-1.96\) and \(+1.96\).
    • Left-tailed will have one critical value negative or at the smaller end; right-tailed positive or greater.
  • Role: Offers a boundary, signaling when observed data contradicts the null hypothesis strongly enough not to dismiss or accept.
By comparing the test statistic with the critical value, we decide to either reject the null hypothesis or not, thereby determining the significance of our test result.

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Most popular questions from this chapter

Consider the null hypothesis \(H_{0}: \mu=5 .\) A random sample of 140 observations is taken from a population with \(\sigma=17\). Using \(\alpha=.05\), show the rejection and nonrejection regions on the sampling distribution curve of the sample mean and find the critical value(s) of \(z\) for the following. \(\begin{array}{lll}\text { a. a right-tailed test } & \text { b. a left-tailed test } & \text { c. a two-tailed test }\end{array}\)

Brooklyn Corporation manufactures DVDs. The machine that is used to make these DVDs is known. to produce not more than \(5 \%\) defective DVDs. The quality control inspector selects a sample of \(200 \mathrm{DVDs}\) each week and inspects them for being good or defective. Using the sample proportion, the quality con trol inspector tests the null hypothesis \(p \leq .05\) against the alternative hypothesis \(p>.05\), where \(p\) is th proportion of DVDs that are defective. She always uses a \(2.5 \%\) significance level. If the null hypothesi: is rejected, the production process is stopped to make any necessary adjustments. A recent sample of 200 DVDs contained 17 defective DVD: Using a \(2.5 \%\) significance level, would you conclude that the prod" should be stoppe to make necessary adjustments b. Perform the test of part a using a \(1 \%\) significance level. Is your decision different from the one in.par Comment on the results of parts a and \(b\)

A random sample of 80 observations produced a sample mean of \(86.50 .\) Find the critical and observed values of \(z\) for each of the following tests of hypothesis using \(\alpha=.10 .\) The population standard deviation is known to be \(7.20\). a. \(H_{0}: \mu=91 \quad\) versus \(\quad H_{1}: \mu \neq 91\) b. \(H_{0}=\mu=91\) versus \(\quad H_{1}: \mu<91\)

A consumer advocacy group suspects that a local supermarket's 10 -ounce packages of cheddar cheese actually weigh less than 10 ounces. The group took a random sample of 20 such packages and found that the mean weight for the sample was \(9.955\) ounces. The population follows a normal distribution with the population standard deviation of \(.15\) ounce. a. Find the \(p\) -value for the test of hypothesis with the alternative hypothesis that the mean weight of all such packages is less than 10 ounces. Will you reject the null hypothesis at \(\alpha=.01\) ? b. Test the hypothesis of part a using the critical-value approach and \(\alpha=.01\).

Consider \(H_{0^{\circ}} \mu=45\) versus \(H_{1}: \mu<45\). a. A random sample of 25 observations produced a sample mean of \(41.8 .\) Using \(\alpha=.025\), would you reject the null hypothesis? The population is known to be normally distributed with \(\sigma=6 .\) b. Another random sample of 25 observations taken from the same population produced a sample mean of \(43.8\). Using \(\alpha=.025\), would you reject the null hypothesis? The population is known to be normally distributed with \(\sigma=6\). Comment on the results of parts a and \(\mathrm{b}\).
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