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According to the National Survey of Student Engagement (NSSE), the average amount of time that first-year college st udents spent preparing for class (studying, reading, writing, doing homework or lab work, analyzing data, rehearsing, and other academic activities) in 2019 was 14.44 hours per week. Your college wonders if the average \(\mu\) for its first-year st udents in 2019 differed from the national average. A random sample of 500 students who were first-year students in 2019 claims to have spent an average of \(x=13.4\) hours per week on homework in their first year. What are the null and alternative hypotheses for a comparison of first-year students at your college with national first-year students in \(2019 ?\) a. \(H_{0}: x=14.44, H_{a}: x \neq 14.44\) b. \(H_{0}: x=13.4, H_{\mathrm{a}}: x>13.4\) c. \(H_{0}: \mu=14.44, H_{a}: \mu \neq 14.44\) d. \(H_{0}: \mu=13.4, H_{a}: \mu>13.4\) Testing Blood Cholesterol. The distribution of blood cholesterol level in the population of all adult patients tested in a large hospital over a 10-year period is close to Normal with mean 130 milligrams per deciliter (mg/dL) and standard deviation \(40 \mathrm{mg} /\) dL. You measure the blood cholesterol of 16 adult patients 20 34 years of age. The mean level is \(x=125 \mathrm{mg} / d \mathrm{~L} . A \mathrm{ssume}\) that \(\sigma\) is the same as in the general hospital populationt. Use this information to artswer Quentions \(19.32\) through \(19.34\)

Step by step solution

01

Identify the Key Information

We know that the national average amount of time that first-year college students spent preparing for class in 2019 was 14.44 hours per week. We also have data from a sample of 500 students from a specific college showing an average of 13.4 hours per week. We need to determine if there's a significant difference in the average time spent.

02

Set Up Null and Alternative Hypotheses

The null hypothesis (\(H_0\)) is that there is no difference between the college students' average and the national average. The alternative hypothesis (\(H_a\)) is that there is a difference. Thus, the null hypothesis is \(H_0: \mu = 14.44\), where \(\mu\) is the population mean for the college. The alternative hypothesis is \(H_a: \mu eq 14.44\).

03

Match Hypotheses to Answer Choices

Now we match our hypotheses to the given choices: - Option (a) states: \(H_{0}: x=14.44, H_{a}: x eq 14.44\), uses sample data rather than population data.- Option (c) states: \(H_{0}: \mu=14.44, H_{a}: \mu eq 14.44\), which matches our formulation and refers to the population mean.Thus, the correct answer is option (c).

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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, denoted as \( H_0 \), represents the default or initial assumption that there is no effect or difference in a particular situation. It serves as the foundation for statistical tests and is assumed true until evidence suggests otherwise. In our original exercise, the null hypothesis posits that the average amount of time ( \( \mu \)) first-year students at your college spent preparing for class equals the known national average of 14.44 hours per week. This assumption forms the basis of our hypothesis test to identify if a significant deviation exists from this average.
To further understand this, keep these points in mind:

• The null hypothesis often involves equality, such as \( \mu = 14.44 \).
• It's a critical tool for determining if observed data can be explained by chance alone.
• Rejecting the null hypothesis under a statistical test implies there is enough evidence to suggest a difference from what's expected.

Alternative Hypothesis

The Alternative Hypothesis, denoted as \( H_a \) or \( H_1 \), is the statement we seek to provide evidence for in hypothesis testing. Unlike the null hypothesis, the alternative hypothesis suggests that there is a difference or effect. In our scenario, the alternative hypothesis states that the average time \( \mu \) spent by first-year students at your college differs from the national average of 14.44 hours. This is expressed as \( H_a: \mu eq 14.44 \).
It's important to consider:

• The alternative hypothesis is not directly tested; rather, it's supported if the null hypothesis is rejected.
• Being a two-tailed test in this case, it covers both possibilities of the average being greater than or less than 14.44 hours.
• Formulating this hypothesis correctly is key to understanding the scope and direction of your research question.

Sample Mean

The Sample Mean, represented by \( x \), is a measure of central tendency that calculates the average from a specific sample data set. In the context of our exercise, the sample mean is 13.4 hours per week, based on the data gathered from 500 first-year students at your college. This value serves as an estimate of the population mean \( \mu \) and is crucial in hypothesis testing to determine if sample data significantly deviates from a known population mean.
Key aspects to remember include:

• The sample mean is used to make inferences about the population mean.
• It reduces complex data into a single value that represents the dataset's central point.
• To draw valuable conclusions, the sample needs to be large enough to represent the population accurately.

Population Mean

The Population Mean, denoted by \( \mu \), is the average of a set of values for the entire population. In statistical analysis, it represents the actual or true mean that we wish to estimate or test against. In our example, the population mean of interest is 14.44 hours, which is the national average for first-year college students based on the survey data.
Some important points about the population mean include:

• It is often unknown and needs to be estimated from sample data.
• Hypothesis testing helps us check if our sample mean accurately reflects or differs significantly from this value.
• Understanding the population mean's connection to sample data is essential for making accurate statistical inferences.