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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 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
For each conjecture: a. H鈧�: \\( \bar{x} \geq 27 \\), H鈧�: \\( \bar{x} < 27 \\) b. H鈧�: \\( \bar{x} = 4.71 \\), H鈧�: \\( \bar{x} \neq 4.71 \\) c. H鈧�: \\( \bar{x} = 36 \\), H鈧�: \\( \bar{x} > 36 \\) d. H鈧�: \\( \bar{x} = 79.95 \\), H鈧�: \\( \bar{x} \neq 79.95 \\) e. H鈧�: \\( \bar{x} = 8.2 \\), H鈧�: \\( \bar{x} \neq 8.2 \\)

Step by step solution

01

Conjecture Analysis

To state the null and alternative hypotheses, we first need to identify the conjecture. For each part of the exercise, the conjecture involves a statement about the average of a population (mean). This is a known parameter (e.g., average age, average experience).
02

State Null Hypothesis (H鈧�)

The null hypothesis is typically a statement of no effect or no difference. For each conjecture, the null hypothesis will state that the population mean is equal to a specific value (or greater for one-sided tests depending on the conjecture).
03

State Alternative Hypothesis (H鈧�)

The alternative hypothesis is what you wish to support, indicating a potential effect or difference from the null hypothesis. It is often a statement of inequality or change from the value stated in the null hypothesis.
04

Formulate Hypotheses for Exercise Parts

a. **Conjecture**: The average age of first-year medical school students is at least 27 years. - **Null Hypothesis (H鈧�)**: \( ar{x} \geq 27 \) - **Alternative Hypothesis (H鈧�)**: \( ar{x} < 27 \)b. **Conjecture**: The average experience (in seasons) for an NBA player is 4.71. - **Null Hypothesis (H鈧�)**: \( ar{x} = 4.71 \) - **Alternative Hypothesis (H鈧�)**: \( ar{x} eq 4.71 \) c. **Conjecture**: The average number of monthly visits/sessions on the Internet by a person at home has increased from 36 in 2009. - **Null Hypothesis (H鈧�)**: \( ar{x} = 36 \) - **Alternative Hypothesis (H鈧�)**: \( ar{x} > 36 \) d. **Conjecture**: The average cost of a cell phone is \\(79.95\\). - **Null Hypothesis (H鈧�)**: \( ar{x} = 79.95 \) - **Alternative Hypothesis (H鈧�)**: \( ar{x} eq 79.95 \) e. **Conjecture**: 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鈧�)**: \( ar{x} = 8.2 \) - **Alternative Hypothesis (H鈧�)**: \( ar{x} eq 8.2 \)

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Key Concepts

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

Null Hypothesis
In hypothesis testing, the null hypothesis (symbolized as \( H_0 \)) is a fundamental concept. It serves as a default or starting assumption about a population parameter. The null hypothesis states that there is no effect or no difference, implying that any observed variation is due to random chance.
For example, if we conjecture that the average age of first-year medical students is at least 27 years, the null hypothesis would be that the mean age is 27 or more. Just as in this exercise, each conjecture you encounter states a certain expected average which becomes the null hypothesis:
  • For the NBA players' experience, \( H_0 : \bar{x} = 4.71 \).
  • For average internet sessions, \( H_0 : \bar{x} = 36 \).
  • For average cell phone cost, \( H_0 : \bar{x} = 79.95 \).
  • For weight loss from exercise, \( H_0 : \bar{x} = 8.2 \).
We test the null hypothesis to determine if it can be rejected in favor of an alternative hypothesis.
Alternative Hypothesis
The alternative hypothesis, denoted as \( H_1 \) or \( H_a \), represents the hypothesis you are trying to support or prove. It suggests that there is a significant effect or difference from what is stated in the null hypothesis.
It poses a direct contrast to \( H_0 \), and you aim to gather evidence against \( H_0 \) to support \( H_1 \):
  • For the age of first-year medical students, the alternative hypothesis is \( H_1 : \bar{x} < 27 \).
  • For the NBA players' experience, \( H_1 : \bar{x} eq 4.71 \), indicating any difference.
  • For internet sessions per month, \( H_1 : \bar{x} > 36 \).
  • For the cost of a cell phone, \( H_1 : \bar{x} eq 79.95 \).
  • For weight loss due to exercise, \( H_1 : \bar{x} eq 8.2 \).
The alternative hypothesis suggests that there is a change or difference, driving the testing process forward.
Population Mean
The population mean is a central concept in hypothesis testing, representing the average of a set of data that reflects an entire population. Often denoted by the Greek letter \( \mu \) (mu), the population mean is a fixed, although often unknown, parameter.
Hypothesis testing frequently revolves around the questioning or estimation of this parameter using sample data. In the context of this exercise, each conjecture centers around a known or hypothesized population mean. For example:
  • The conjecture that medical students' average age is at least 27.
  • The average experience of NBA players at 4.71 seasons.
  • The mean monthly internet visits at home being 36 in 2009.
  • The cost of a cell phone being \$79.95.
  • The average weight loss being 8.2 pounds with exercise.
Each of these conjectures uses the population mean as a basis for hypothesis testing to understand whether it holds true under statistical scrutiny.
Statistical Inference
Statistical inference is the broader concept underpinning the process of making conclusions about a population based on sample data. It involves determining properties of an underlying distribution using statistical methods.
In hypothesis testing, statistical inference allows us to make educated decisions or predictions. By testing the null and alternative hypotheses, you infer whether the data supports or contradicts your initial assumptions. This process involves several key steps:
  • Collecting sample data that you expect to represent the population.
  • Formulating null and alternative hypotheses based on your conjecture.
  • Using statistical tests to calculate the probability of observing your data if the null hypothesis is true.
  • Making a decision to reject or fail to reject the null hypothesis based on this probability.
Statistical inference helps bridge the gap between sample data and population insights, providing a mathematical basis for determining significance in your observed data.

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Most popular questions from this chapter

A manager states that in his factory, the average number of days per year missed by the employees due to illness is less than the national average of \(10 .\) The following data show the number of days missed by 40 randomly selected employees last year. Is there sufficient evidence to believe the manager's statement at \(\alpha=0.05 ? \sigma=3.63 .\) Use the \(P\) -value method. \(\begin{array}{rrrrrrrr}0 & 6 & 12 & 3 & 3 & 5 & 4 & 1 \\ 3 & 9 & 6 & 0 & 7 & 6 & 3 & 4 \\ 7 & 4 & 7 & 1 & 0 & 8 & 12 & 3 \\ 2 & 5 & 10 & 5 & 15 & 3 & 2 & 5 \\ 3 & 11 & 8 & 2 & 2 & 4 & 1 & 9\end{array}\)

Ten years ago, the average acreage of farms in a certain geographic region was 65 acres. The standard deviation of the population was 7 acres. A recent study consisting of 22 randomly selected farms showed that the average was 63.2 acres per farm. Test the claim, at \(\alpha=0.10,\) that the average has not changed by finding the \(P\) -value for the test. Assume that \(\sigma\) has not changed and the variable is normally distributed.

The average depth of the Hudson Bay is 305 feet. Climatologists were interested in seeing if warming and ice melt were affecting the water level. Fifty-five measurements over a period of randomly selected weeks yielded a sample mean of 306.2 feet. The population variance is known to be \(3.6 .\) Can it be concluded at the 0.05 level of significance that the average depth has increased? Is there evidence of what caused this to happen?

Perform each of the following steps. a. State the hypotheses and identify the claim. b. Find the critical value(s). c. Compute the test value. d. Make the decision. e. Summarize the results. Use the traditional method of hypothesis testing unless otherwise specified. Researchers suspect that \(18 \%\) of all high school students smoke at least one pack of cigarettes a day. At Wilson High School, a randomly selected sample of 300 students found that 50 students smoked at least one pack of cigarettes a day. At \(\alpha=0.05,\) test the claim that less than \(18 \%\) of all high school students smoke at least one pack of cigarettes a day. Use the \(P\) -value method.

According to the almanac, the mean age for a woman giving birth for the first time is 25.2 years. A random sample of ages of 35 professional women giving birth for the first time had a mean of 28.7 years and a standard deviation of 4.6 years. Use both a confidence interval and a hypothesis test at the 0.05 level of significance to test if the mean age of professional woman is different from 25.2 years at the time of their first birth.
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