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In statistics, a null hypothesis serves as the starting point for conducting a hypothesis test. It is essentially a statement, typically about a population parameter, formulated to be tested.
In our problem, the service station manager claims that the mean amount spent on gas by customers is \(15.90. Thus, the null hypothesis, denoted as \(H_0\), asserts that the mean spending is indeed \)15.90.
This can be mathematically expressed as:
\(H_0: \mu = 15.90\).
- The null hypothesis is always about population parameters, not sample statistics.- It often represents a "no effect" or "status quo" situation. - The goal of the hypothesis test is to assess the evidence against the null hypothesis.

The alternative hypothesis offers a statement that contradicts the null hypothesis. It suggests a different scenario than the one proposed under the null. In our example, you want to test whether the mean amount spent on gas is different from \(15.90. Thus, the alternative hypothesis, denoted as \(H_a\), posits that the mean is not equal to \)15.90:
\(H_a: \mu eq 15.90\).
- The alternative hypothesis reflects what we seek to support with our test.- It can be one-sided (greater than or less than a certain value) or two-sided (not equal to a specific value).

In this case, it is a two-sided hypothesis, implying we are interested in any deviation from $15.90, not just an increase or decrease.

The test statistic is a crucial component of hypothesis testing. It is calculated using sample data and compared against a critical value to decide whether to reject the null hypothesis.
When you're using a t-test, the formula for the test statistic \( t \) is:
\[ t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}} \] where:- \( \bar{x} \) is the sample mean- \( \mu_0 \) is the population mean under the null hypothesis (e.g., $15.90)- \( s \) is the sample standard deviation- \( n \) is the sample size

• A t-test is particularly useful when the population standard deviation is unknown.
• The calculated t-score indicates how far and in what direction the sample mean deviates from the null hypothesis mean, in units of standard error.

This comparison helps determine if the observed data is significantly different from what was stated in the null hypothesis.

A critical value establishes the threshold for deciding whether to reject the null hypothesis. It is linked to the confidence level you choose for your test, typically 95% or 99%. In a t-test, the critical value is determined based on the degrees of freedom, which are calculated as \( n - 1 \), where \( n \) is the sample size.
- Look up the critical t-value from a t-distribution table or use statistical software.- Compare the absolute value of the calculated test statistic to this critical value.If the test statistic exceeds the critical value, the null hypothesis is rejected. This suggests the sample provides enough evidence to conclude that the mean amount spent on gas is indeed different from $15.90. On the other hand, if the statistic is less than the critical value, we do not reject the null hypothesis, indicating there is not enough evidence to support a difference. This approach forms the basis of making decisions in hypothesis testing.