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Chapter 9: Problem 64

A statistics instructor believes that fewer than 20% of Evergreen Valley College (EVC) students attended the opening night midnight showing of the latest Harry Potter movie. She surveys 84 of her students and finds that 11 attended the midnight showing. An appropriate alternative hypothesis is: a. \(p=0.20\) b. \(p > 0.20\) c. \(p < 0.20\) d. \(p \leq 0.20\)

Short Answer

Expert verified
The appropriate alternative hypothesis is: c. \(p < 0.20\).

Step by step solution

01

Understand the Problem Statement

The statistics instructor suspects that fewer than 20% of the students attended the midnight showing of the Harry Potter movie. In hypothesis testing, this type of statement leads us to form a hypothesis where the proportion is less than a certain value.
02

Identify the Null and Alternative Hypothesis

The null hypothesis (H_0) generally represents a statement of no effect or no difference, and it is typically the complement of the alternative hypothesis. The alternative hypothesis (H_a) is the claim the instructor is testing. Here, the instructor's claim is that the proportion of students attending is less than 20%. Thus, we have:\(H_0: p = 0.20\) \(H_a: p < 0.20\)
03

Select the Appropriate Alternative Hypothesis

The given options are potential alternative hypotheses. Since the instructor believes fewer than 20% attended, the option that aligns with this notion is the one where the proportion is less than 0.20. Therefore, the correct alternative hypothesis is: \(p < 0.20\). This corresponds to option c.

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Key Concepts

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

Alternative Hypothesis
When conducting a hypothesis test, the alternative hypothesis plays a central role. It's the statement that reflects the researcher's actual belief or suspicion. In our problem, the instructor suspects fewer than 20% of students went to a movie premiere. The alternative hypothesis, denoted as \(H_a\), captures this claim mathematically. Here, \(H_a: p < 0.20\), clearly indicates the instructor's belief that the proportion of students who attended is less than 20%. This is a one-tailed hypothesis test where we focus on values that are only less than the specified proportion of 0.20.
  • It challenges the null hypothesis by asserting a statistically significant difference.
  • In hypothesis testing, we gather data to support the alternative hypothesis, ultimately leading to its acceptance or rejection.
  • It is often what the analyst aims to provide evidence for, thus making it the primary hypothesis of interest.
Null Hypothesis
The null hypothesis serves as the default or starting assumption in hypothesis testing. Represented as \(H_0\), it is a statement of no change, no difference, or no effect. For the instructor's study, the null hypothesis is \(H_0: p = 0.20\). This states that exactly 20% of the students attended the midnight movie showing.
  • The null hypothesis is essential because it provides a specific statement to test against, often suggesting no effect or association.
  • It is always assumed true until evidence suggests otherwise, which helps draw objective conclusions from the statistical test.
  • Rejecting the null hypothesis gives a basis to accept the alternative hypothesis with statistical evidence backing the claim.
In statistical tests, the decision to reject or not reject the null hypothesis is informed by the p-value or test statistic calculated from the data.
Proportion Hypothesis
A proportion hypothesis relates directly to claims involving proportions or percentages within a population. In this scenario, we are examining the proportion of students who saw the Harry Potter film premiere. Proportion hypotheses are often used when we're interested in the fraction of the population displaying a specific attribute or outcome. For example, the hypothesis involves testing whether the true proportion, \(p\), is less than 20%.
  • These hypotheses are typically framed in relation to a certain benchmark or statistical baseline, such as 0.20 in this example.
  • To test a proportion hypothesis, one typically calculates the sample proportion and compares it to the hypothesized proportion using statistical tests like the z-test for proportions.
  • Assessing the proportion helps inform whether the population parameter significantly differs from the hypothesized value.
Understanding and utilizing proportion hypotheses allow us to make informed decisions based on statistical examinations of sample data.

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Most popular questions from this chapter

The mean entry level salary of an employee at a company is $58,000. You believe it is higher for IT professionals in the company. State the null and alternative hypotheses.

Previously, an organization reported that teenagers spent 4.5 hours per week, on average, on the phone. The organization thinks that, currently, the mean is higher. Fifteen randomly chosen teenagers were asked how many hours per week they spend on the phone. The sample mean was 4.75 hours with a sample standard deviation of 2.0. Conduct a hypothesis test. At a significance level of \(a=0.05,\) what is the correct conclusion? a. There is enough evidence to conclude that the mumber of hours is more than 4.75 b. There is enough evidence to conclude that the mean number of hore than 4.5 c. There is not enough evidence to conclude that the mumber of hours is more than 4.5 d. There is not enough evidence to conclude that the mean number of hours is more than 4.75

According to an article in Bloomberg Businessweek, New York City's most recent adult smoking rate is 14%. Suppose that a survey is conducted to determine this year鈥檚 rate. Nine out of 70 randomly chosen N.Y. City residents reply that they smoke. Conduct a hypothesis test to determine if the rate is still 14% or if it has decreased.

The mean age of graduate students at a University is at most 31 y ears with a standard deviation of two years. A random sample of 15 graduate students is taken. The sample mean is 32 years and the sample standard deviation is three years. Are the data significant at the 1\(\%\) level? The p-value is \(0.0264 .\) State the null and alternative hypotheses and interpret the \(p\) -value.

"Dalmatian Darnation," by Kathy Sparling A greedy dog breeder named Spreckles Bred puppies with numerous freckles The Dalmatians he sought Possessed spot upon spot The more spots, he thought, the more shekels. His competitors did not agree That freckles would increase the fee. They said, 鈥淪pots are quite nice But they don't affect price; One should breed for improved pedigree.鈥� The breeders decided to prove This strategy was a wrong move. Breeding only for spots Would wreak havoc, they thought. His theory they want to disprove. They proposed a contest to Spreckles Comparing dog prices to freckles. In records they looked up One hundred one pups: Dalmatians that fetched the most shekels. They asked Mr. Spreckles to name An average spot count he'd claim To bring in big bucks. Said Spreckles, 鈥淲ell, shucks, It's for one hundred one that I aim.鈥� Said an amateur statistician Who wanted to help with this mission. 鈥淭wenty-one for the sample Standard deviation's ample: They examined one hundred and one Dalmatians that fetched a good sum. They counted each spot, Mark, freckle and dot And tallied up every one. Instead of one hundred one spots They averaged ninety six dots Can they muzzle Spreckles鈥� Obsession with freckles Based on all the dog data they've got?
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