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Chapter 8: Problem 14

For each conjecture, state the null and alternative hypotheses. a. The average age of first-year medical school students is at least 27 years. b. The average experience (in seasons) for an \(\mathrm{NBA}\) player is 4.71 . c. The average number of monthly visits/sessions on the Internet by a person at home has increased from 36 in 2009 . d. The average cost of a cell phone is \(\$ 79.95 .\) e. The average weight loss for a sample of people who exercise 30 minutes per day for 6 weeks is 8.2 pounds.

Short Answer

Expert verified
a) H_0: \( \mu \geq 27 \), H_1: \( \mu < 27 \). b) H_0: \( \mu = 4.71 \), H_1: \( \mu \neq 4.71 \). c) H_0: \( \mu \leq 36 \), H_1: \( \mu > 36 \). d) H_0: \( \mu = 79.95 \), H_1: \( \mu \neq 79.95 \). e) H_0: \( \mu = 8.2 \), H_1: \( \mu \neq 8.2 \).

Step by step solution

01

Understanding Hypotheses

In hypothesis testing, the **Null Hypothesis ( H_0 d)** is a statement of no effect or status quo, while the **Alternative Hypothesis ( H_1 d)** indicates the presence of an effect or difference. We use statistical tests to determine whether to reject H_0 in favor of H_1 .
02

Formulating Hypotheses for Part a

**Conjecture a:** "The average age of first-year medical school students is at least 27 years." - Null Hypothesis (H_0d): the average age is 27 years or more, stated as H_0: \( \mu \geq 27 \).- Alternative Hypothesis (H_1d): the average age is less than 27 years, stated as H_1: \( \mu < 27 \).
03

Formulating Hypotheses for Part b

**Conjecture b:** "The average experience (in seasons) for an NBA player is 4.71." - Null Hypothesis (H_0d): the average experience is 4.71 seasons, stated as H_0: \( \mu = 4.71 \).- Alternative Hypothesis (H_1d): the average experience is not 4.71 seasons, stated as H_1: \( \mu eq 4.71 \).
04

Formulating Hypotheses for Part c

**Conjecture c:** "The average number of monthly visits/sessions on the Internet by a person at home has increased from 36 in 2009." - Null Hypothesis (H_0d): the average number of visits is 36 or less, stated as H_0: \( \mu \leq 36 \).- Alternative Hypothesis (H_1d): the average number of visits is more than 36, stated as H_1: \( \mu > 36 \).
05

Formulating Hypotheses for Part d

**Conjecture d:** "The average cost of a cell phone is \(79.95." - Null Hypothesis (H_0d): the average cost is \)79.95, stated as H_0: \( \mu = 79.95 \).- Alternative Hypothesis (H_1d): the average cost is not $79.95, stated as H_1: \( \mu eq 79.95 \).
06

Formulating Hypotheses for Part e

**Conjecture e:** "The average weight loss for a sample of people who exercise 30 minutes per day for 6 weeks is 8.2 pounds." - Null Hypothesis (H_0d): the average weight loss is 8.2 pounds, stated as H_0: \( \mu = 8.2 \).- Alternative Hypothesis (H_1d): the average weight loss is not 8.2 pounds, stated as H_1: \( \mu eq 8.2 \).

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Key Concepts

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

Null Hypothesis
The null hypothesis, represented as \( H_0 \), is a foundational concept in hypothesis testing. It serves as a "default" or "starting" assumption. Essentially, the null hypothesis proposes that there is no significant effect or difference in the context of our problem. In other words, it typically suggests that any observed variations are due to chance or natural fluctuations rather than a specific cause.
  • For Conjecture a: The null hypothesis is that the average age of first-year medical students is at least 27 years. Formulated as \( \mu \geq 27 \).
  • For Conjecture b: It states the average NBA player experience is exactly 4.71 seasons, written as \( \mu = 4.71 \).
  • For Conjecture c: It posits monthly internet visits have not increased from 36, \( \mu \leq 36 \).
  • For Conjecture d: The average cost of a cell phone remains \( \$79.95 \), \( \mu = 79.95 \).
  • For Conjecture e: Average weight loss is strictly 8.2 pounds, expressed as \( \mu = 8.2 \).
The null hypothesis is crucial because it sets the stage for statistical tests. These tests help determine if this baseline assumption can be confidently rejected.
Alternative Hypothesis
The alternative hypothesis, represented as \( H_1 \), is a statement that contradicts the null hypothesis. It suggests the presence of an effect or difference. By testing the null hypothesis, we indirectly assess the likelihood of the alternative hypothesis being true.
  • For Conjecture a: This posits that the average age is less than 27 years; formulated as \( \mu < 27 \).
  • For Conjecture b: Here, the suggestion is that the average experience differs from 4.71, giving us \( \mu eq 4.71 \).
  • For Conjecture c: It proposes an increase in monthly internet visits exceeding 36, expressed by \( \mu > 36 \).
  • For Conjecture d: It claims the average cell phone cost isn't exactly \( \$79.95 \), presented as \( \mu eq 79.95 \).
  • For Conjecture e: It indicates average weight loss diverges from 8.2 pounds, stated as \( \mu eq 8.2 \).
Each alternative hypothesis usage offers a pathway to explore genuine differences or changes, encouraging deeper scientific inquiry.
Statistical Tests
When engaging in hypothesis testing, statistical tests are the tools used to decide whether to reject the null hypothesis. These tests assess the probability that the observed data could have occurred by chance under the null hypothesis.
  • Common Tests: Tests such as the t-test or chi-square test are frequently employed, based on the data type and distribution.
  • Decision Making: The main goal is to ascertain if the observed effect is statistically significant, i.e., unlikely to have occurred due to random chance.
  • Significance Level: This is often set at 0.05, meaning there's a 5% risk of concluding that a difference exists when there is none.
These tests provide a systematic approach to backing decisions in research and are central to supporting claims drawn from statistical analysis.
Average Hypothesis
The average hypothesis relates to the specific conjecture concerning the mean value of a dataset. It is a type of hypothesis that concerns expected values, such as means or averages, that one aims to compare. Reasons to Focus on Averages:
  • Comparability: Averages make it easier to compare datasets effectively.
  • Reduction of complexity: They distill data into a single representative value, simplifying analysis.
In the context of the exercise, each conjecture deals with average figures, whether it's age, experience, costs, or weight loss, hence focusing on average hypotheses provides insights into these typical values and their implications in real-world terms. Such hypotheses are central themes in many experiments and help to streamline otherwise complex datasets.

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Most popular questions from this chapter

For Exercises 5 through \(20,\) assume that the variables are normally or approximately normally distributed. Use the traditional method of hypothesis testing unless otherwise specified. High Temperatures in January Daily weather observations for southwestern Pennsylvania for the first three weeks of January for randomly selected years show daily high temperatures as follows: \(55,44,51,59,62,\) \(60,46,51,37,30,46,51,53,57,57,39,28,37,35,\) and 28 degrees Fahrenheit. The normal standard deviation in high temperatures for this time period is usually no more than 8 degrees. A meteorologist believes that with the unusual trend in temperatures the standard deviation is greater. At \(\alpha=0.05,\) can we conclude that the standard deviation is greater than 8 degrees?

In hypothesis testing, why can't the hypothesis be proved true?

For Exercises 7 through \(23,\) perform each of the following steps. a. State the hypotheses and identify the claim. b. Find the critical value(s). c. Find the test value. d. Make the decision. e. Summarize the results. Use the traditional method of hypothesis testing unless otherwise specified. Assume that the population is approximately normally distributed. Cell Phone Call Lengths The average local cell phone call length was reported to be 2.27 minutes. A random sample of 20 phone calls showed an average of 2.98 minutes in length with a standard deviation of 0.98 minute. At \(\alpha=0.05,\) can it be concluded that the average differs from the population average?

Newspaper Reading Times A survey taken several years ago found that the average time a person spent reading the local daily newspaper was 10.8 minutes. The standard deviation of the population was 3 minutes. To see whether the average time had changed since the newspaper's format was revised, the newspaper editor surveyed 36 individuals. The average time that these 36 randomly selected people spent reading the paper was 12.2 minutes. At $$\alpha=0.02,$$ is there a change in the average time an individual spends reading the newspaper? Find the $$98 \%$$ confidence interval of the mean. Do the results agree? Explain.

For Exercises 7 through \(23,\) perform each of the following steps. a. State the hypotheses and identify the claim. b. Find the critical value(s). c. Find the test value. d. Make the decision. e. Summarize the results. Use the traditional method of hypothesis testing unless otherwise specified. Assume that the population is approximately normally distributed. Number of Jobs The U.S. Bureau of Labor and Statistics reported that a person between the ages of 18 and 34 has had an average of 9.2 jobs. To see if this average is correct, a researcher selected a random sample of 8 workers between the ages of 18 and 34 and asked how many different places they had worked. The results were as follows: $$\begin{array}{ccccccccc}{8} & {12} & {15} & {6} & {1} & {9} & {13} & {2}\end{array}$$ At \(\alpha=0.05,\) can it be concluded that the mean is \(9.2 ?\) Use the \(P\) -value method. Give one reason why the respondents might not have given the exact number of jobs that they have worked.
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