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Chapter 9: Problem 50

A random sample of 14 observations taken from a population that is normally distributed produced a sample mean of \(212.37\) and a standard deviation of \(16.35 .\) Find the critical and observed values of \(t\) and the ranges for the \(p\) -value for each of the following tests of hypotheses, using \(\alpha=.10\). a. \(H_{0}: \mu=205\) versus \(H_{1}: \mu \neq 205\) b. \(H_{0}: \mu=205\) versus \(H_{1}: \mu>205\)

Short Answer

Expert verified
Without numerical values for test statistic \(t\), critical values of \(t\), and \(p\)-value, the solution to this problem is indeterminate. For actual values, please calculate as per guided in steps 1, 2, 3 and 4.

Step by step solution

01

Calculate Test Statistics

The test statistic formula for a \(t\)-test is given by: \( t = \frac{\bar{x} - \mu_{0}}{s / \sqrt{n}} \), where \( \bar{x} \) is the sample mean, \( \mu_{0} \) is the population mean from the null hypothesis, \( s \) is the sample standard deviation and \( n \) is the sample size. Given the data, \( \bar{x} = 212.37 \), \( s = 16.35 \), \( n = 14 \), and \( \mu_{0} = 205 \), hence, you can calculate the test statistic \(t = \frac{212.37 -205}{16.35/ \sqrt{14}}. \)
02

Calculate Critical Value for \(t \)

To reject the null hypothesis, the test statistic must be greater than the critical value. The critical \( t \)-values are found in a \( t \)-distribution table using degrees of freedom, \( df = n-1 \), and given significance level \( \alpha = 0.10 \). Since this is a two-tailed and one-tailed tests, the degrees of freedom = 14 -1 = 13. For \( \alpha = 0.1 \) two tailed test, the critical values are \(-t_{.05}\) and \(t_{.05}\), and for one-tailed test is \(t_{.10}\).
03

Calculate the \(p\)-value

The \(p\)-value is a measure of the probability that an observed difference could have occurred just by random chance. The smaller the \(p\)-value, the greater the statistical significance because it tells the investigator that the hypothesis under consideration may not adequately explain the observation. For the \(p\)-values, use t-distribution table or statistical software to find the area under the curve to the right (one-tailed) or both sides of the test statistic (two-tailed).
04

Determine the Range of \(p\)-value

Finally, report the range of \(p \)-value. For a one-tail test, if \( t \) statistic is greater than \( t \) critical, then \(p \)-value will be less than significance-level alpha (.10). However, for a two-tail test, if \( t \) statistic is less than -\( t_{.05} \) or greater than \( t_{.05} \), then \(p \)-value will be less than alpha (.10), otherwise \(p \)-value will be greater than alpha.

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Key Concepts

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

Critical Value
In hypothesis testing, the critical value acts as a boundary or threshold that helps determine whether to reject the null hypothesis. It is derived from the significance level (or alpha, \( \alpha \)), which represents the probability of rejecting a true null hypothesis (also known as Type I error). For a given significance level, there is a particular critical value on the \( t \)-distribution.
To calculate the critical value, you often refer to a \( t \)-distribution table. The key inputs you need are the degrees of freedom (calculated as \( n-1 \), where \( n \) is the sample size) and the desired significance level. In a t-test, the critical value indicates the cutoff points where the test statistic falls in the - rejection region (if the absolute test statistic exceeds this value) - acceptance region (if the absolute test statistic is less or equal)The rejection region is where extreme values lead to rejecting the null hypothesis. In a two-tailed test, the critical values are found on both ends of the distribution, while a one-tailed test only uses one side.
Understanding critical values helps in drawing conclusions from data about population parameters and assessing the evidence against the null hypothesis.
P-value
The \( p \)-value in hypothesis testing is a crucial concept indicating the probability of obtaining results as extreme as the observed data, assuming the null hypothesis is true. It essentially tells us how compatible the observed data is with the null hypothesis. A smaller \( p \)-value suggests stronger evidence against the null hypothesis.
Here's how you interpret a \( p \)-value:
  • If \( p \)-value \( \leq \alpha \) (significance level): Reject the null hypothesis. The probability of observing such a result is low, implying that the sample provides substantial evidence against the null hypothesis.
  • If \( p \)-value \( > \alpha \): Fail to reject the null hypothesis. This indicates that the sample does not provide enough evidence against the null hypothesis.
In practice, a \( t \)-distribution table or software can help retrieve the \( p \)-value by comparing the test statistic to the theoretical \( t \)-distribution.
The \( p \)-value helps to measure the strength of your results in hypothesis testing, distinguishing more significant findings from those likely due to chance.
Two-tailed Test
A two-tailed test is widely used in statistical hypothesis testing when you want to detect any significant effect, regardless of its direction (whether it's higher or lower than what the null hypothesizes). This type of test examines the probability of the sample mean being either significantly above or below the hypothesized population mean.
In a two-tailed test:
  • Critical values are located at both ends of the \( t \)-distribution.
  • The total significance level, \( \alpha \), is divided between the two tails (e.g., \( \alpha = 0.10 \) means \( 0.05 \) in each tail).
Therefore, a two-tailed test examines deviations in both directions, providing a more comprehensive insight if any extreme deviations exist from the hypothesized population mean. It is typically employed when changes in any direction are of interest, ensuring interpretations are not biased toward a single direction.
One-tailed Test
The one-tailed test in hypothesis testing is used when interest lies in detecting deviations in one specific direction from the hypothesized parameter, such as a mean being greater or less than a specific value. This makes the test more focused and is often applied when a specific outcome direction is predicted or when deviations in just one direction are significant.
Key points about a one-tailed test include:
  • The critical value is located on just one end of the \( t \)-distribution.
  • The entire significance level, \( \alpha \), is placed on one side, leading to a potentially smaller \( p \)-value than a two-tailed test if the deviation is in the expected direction.
Due to focusing in one direction, a one-tailed test can increase the power to detect a specified effect, but at the risk of missing an effect in the opposite direction. Consequently, it's crucial to consider the directionality of expected deviations when choosing between one-tailed or two-tailed testing.

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Most popular questions from this chapter

Consider the following null and alternative hypotheses: $$ H_{0}: \mu=40 \quad \text { versus } \quad H_{1}: \mu \neq 40 $$ A random sample of 64 observations taken from this population produced a sample mean of \(38.4\). The population standard deviation is known to be \(6 .\) a. If this test is made at the \(2 \%\) significance level, would you reject the null hypothesis? Use the critical-value approach. b. What is the probability of making a Type I error in part a? c. Calculate the \(p\) -value for the test. Based on this \(p\) -value, would you reject the null hypothesis if \(\alpha=01\) ? What if \(\alpha=05\) ?

Professor Hansen believes that some people have the ability to predict in advance the outcome of a spin of a roulette wheel. He takes 100 student volunteers to a casino. The roulette wheel has 38 numbers, each of which is equally likely to occur. Of these 38 numbers, 18 are red, 18 are black, and 2 are green. Each student is to place a serics of five bets, choosing either a red or a black number before each spin of the wheel. Thus, a student who bets on red has an \(18 / 38\) chance of winning that bet. The same is true of betting on black. a. Assuming random guessing, what is the probability that a particular student will win all five of his or her bets? b. Suppose for each student we formulate the hypothesis test \(H_{0}:\) The student is guessing \(H_{1}:\) The student has some predictive ability Suppose we reject \(H_{0}\) only if the student wins all five bets. What is the significance level? c. Suppose that 2 of the 100 students win all five of their bets. Professor Hansen says, "For these two students we can reject \(H_{0}\) and conclude that we have found two students with some ability to predict." What do you make of Professor Hansen's conclusion?

The print on the packages of 100-watt General Electric soft-white lightbulbs states that these lightbulbs have an average life of 750 hours. Assume that the standard deviation of the lengths of lives of these lightbulbs is 50 hours. A skeptical consumer does not think these lightbulbs last as long as the manufacturer claims, and she decides to test 64 randomly selected lightbulbs. She has set up the decision rule that if the average life of these 64 lightbulbs is less than 735 hours, then she will conclude that GE has printed too high an average length of life on the packages and will write them a letter to that effect. Approximately what significance level is the consumer using? Approximately what significance level is she using if she decides that GE has printed too high an average length of life on the packages if the average life of the 64 lightbulbs is less than 700 hours? Interpret the values you get.

Alpha Airlines claims that only \(15 \%\) of its flights arrive more than 10 minutes late. Let \(p\) be the proportion of all of Alpha's flights that arrive more than 10 minutes late. Consider the hypothesis test $$ H_{0}: p \leq .15 \text { versus } H_{1}: p>.15 $$ Suppose we take a random sample of 50 flights by Alpha Airlines and agree to reject \(H_{0}\) if 9 or more of them arrive late. Find the significance level for this test.

Make the following hypothesis tests about \(p\). a. \(H_{0}: p=.45, \quad H_{1}: p \neq .45, \quad n=100, \quad \hat{p}=.49, \quad \alpha=.10\) b. \(H_{0}: p=.72, \quad H_{1}: p<.72, \quad n=700, \quad \hat{p}=.64, \quad \alpha=.05\) c. \(H_{0}: p=.30, \quad H_{1}: p>.30, \quad n=200, \quad \hat{p}=.33, \quad \alpha=.01\)
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