It is suggested that the mean one-day travel expense by tourists in Moscow is $500 (assume all one-day travel expenses in Moscow are approximately normally distributed). In a random sample of 16 Moscow tourists, the sample mean one-day travel expense was $482 with a sample standard deviation s = 22.5. Our goal is to test whether or not the mean one-day travel expense is $500 or not.1) What is the null hypotheses? H0: 2) What is the alternative hypotheses? Ha: **Determining the Null Hypothesis:**In the context of hypothesis testing, the null hypothesis ($ H_0 $) is a statement that there is no effect or no difference, and it is the default or starting assumption in scientific research. The critical value and p-value approaches are commonly used to test hypotheses.The following options are given for selecting the correct null hypothesis:- $ p = 482 $- $ \mu = 500 $- $ p = 500 $- $ \mu = 482 $Below this, the question is posed:**"What is the null hypothesis? $ H_0 $:"**### Explanation:1. **$ p $ represents the population proportion**: This symbol is typically used when dealing with categorical data and binomial distributions.2. **$ \mu $ represents the population mean**: This symbol is used for continuous data and normal distributions.Based on the context of the problem (e.g., mean travel expense), if we need to test a population mean, we would select an option with $ \mu $. If we're testing for a population proportion, we would select an option with $ p $.Given these choices:- If the sample relates to the population mean, the options would be either $ \mu = 500 $ or $ \mu = 482 $.- If it's related to the population proportion, the choices would be either $ p = 482 $ or $ p = 500 $.The selection depends on the research context, which in this case seems to relate to a travel expense. If the average travel expense is discussed, likely, the mean ($ \mu $) will be used.Thus, the correct null hypothesis should be framed with $ \mu $ as the symbol.Note: Please ensure that you refer to the specific research question you are investigating to choose the appropriate null hypothesis. ### Understanding Alternative Hypotheses in Hypothesis TestingWhen conducting hypothesis testing in statistics, one needs to set up two competing hypotheses: the null hypothesis (often denoted as $ H_0 $) and the alternative hypothesis (denoted as $ H_a $). The null hypothesis reflects the status quo or a statement of no effect or no difference, while the alternative hypothesis represents what the researcher aims to support.#### Example Question:**What is the alternative hypothesis? $ H_a $:**Below are the options provided:1. $ p < 482 $2. $ p > 500 $3. $ \mu \neq 500 $4. $ \mu = 482 $Each option represents a possible form of an alternative hypothesis:- **$ p < 482 $:** This suggests that the parameter $ p $ is less than 482.- **$ p > 500 $:** This indicates that the parameter $ p $ is greater than 500.- **$ \mu \neq 500 $:** This represents a two-tailed test where the parameter $ \mu $ is not equal to 500.- **$ \mu = 482 $:** This indicates that the parameter $ \mu $ is exactly equal to 482.In hypothesis testing, the alternative hypothesis $ H_a $ is typically set to detect the presence of an effect, a difference, or a relationship. The form of the alternative hypothesis depends on the research question and the nature of the data.In multiple-choice questions, selecting the correct form of the alternative hypothesis is crucial for framing the right type of test to be conducted, whether it's a one-tailed or a two-tailed test.---This information can help students and researchers correctly identify and establish their alternative hypothesis, leading to more accurate and effective testing in their statistical analyses.

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It is suggested that the mean one-day travel expense by tourists in Moscow is $500 (assume all one-day travel expenses in Moscow are approximately normally distributed). In a random sample of 16 Moscow tourists, the sample mean one-day travel expense was $482 with a sample standard deviation s = 22.5. Our goal is to test whether or not the mean one-day travel expense is $500 or not.

1) What is the null hypotheses? H0:

2) What is the alternative hypotheses? Ha:

**Determining the Null Hypothesis:**

In the context of hypothesis testing, the null hypothesis ($ H_0 $) is a statement that there is no effect or no difference, and it is the default or starting assumption in scientific research. The critical value and p-value approaches are commonly used to test hypotheses.

The following options are given for selecting the correct null hypothesis:

- $ p = 482 $
- $ \mu = 500 $
- $ p = 500 $
- $ \mu = 482 $

Below this, the question is posed:
**"What is the null hypothesis? $ H_0 $:"**

### Explanation:

1. **$ p $ represents the population proportion**: This symbol is typically used when dealing with categorical data and binomial distributions.

2. **$ \mu $ represents the population mean**: This symbol is used for continuous data and normal distributions.

Based on the context of the problem (e.g., mean travel expense), if we need to test a population mean, we would select an option with $ \mu $. If we're testing for a population proportion, we would select an option with $ p $.

Given these choices:
- If the sample relates to the population mean, the options would be either $ \mu = 500 $ or $ \mu = 482 $.
- If it's related to the population proportion, the choices would be either $ p = 482 $ or $ p = 500 $.

The selection depends on the research context, which in this case seems to relate to a travel expense. If the average travel expense is discussed, likely, the mean ($ \mu $) will be used.

Thus, the correct null hypothesis should be framed with $ \mu $ as the symbol.

Note: Please ensure that you refer to the specific research question you are investigating to choose the appropriate null hypothesis.

### Understanding Alternative Hypotheses in Hypothesis Testing

When conducting hypothesis testing in statistics, one needs to set up two competing hypotheses: the null hypothesis (often denoted as $ H_0 $) and the alternative hypothesis (denoted as $ H_a $). The null hypothesis reflects the status quo or a statement of no effect or no difference, while the alternative hypothesis represents what the researcher aims to support.

#### Example Question:
**What is the alternative hypothesis? $ H_a $:**

Below are the options provided:

1. $ p < 482 $
2. $ p > 500 $
3. $ \mu \neq 500 $
4. $ \mu = 482 $

Each option represents a possible form of an alternative hypothesis:

- **$ p < 482 $:** This suggests that the parameter $ p $ is less than 482.
- **$ p > 500 $:** This indicates that the parameter $ p $ is greater than 500.
- **$ \mu \neq 500 $:** This represents a two-tailed test where the parameter $ \mu $ is not equal to 500.
- **$ \mu = 482 $:** This indicates that the parameter $ \mu $ is exactly equal to 482.

In hypothesis testing, the alternative hypothesis $ H_a $ is typically set to detect the presence of an effect, a difference, or a relationship. The form of the alternative hypothesis depends on the research question and the nature of the data.

In multiple-choice questions, selecting the correct form of the alternative hypothesis is crucial for framing the right type of test to be conducted, whether it's a one-tailed or a two-tailed test.

---

This information can help students and researchers correctly identify and establish their alternative hypothesis, leading to more accurate and effective testing in their statistical analyses.

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...

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