A manufacturer claims that its new improved tire design...

A manufacturer claims that its new improved tire design increases tire lifetime from $50,000 \mathrm{~km}$ to $55,000 \mathrm{~km}$. A test of 8 tires gives a sample mean lifetime of $52,500 \mathrm{~km}$ and a sample standard deviation of 3000 km . (a) Find a test to determine if the claim can be supported at a level of $\alpha=1 \%$; $\alpha=5 \%$. (b) Find the $p$-value of the observation.

A manufacturer claims that its new improved tire design increases tire lifetime from $50,000 \mathrm{~km}$ to $55,000 \mathrm{~km}$. A test of 8 tires gives a sample mean lifetime of $52,500 \mathrm{~km}$ and a sample standard deviation of 3000 km .
(a) Find a test to determine if the claim can be supported at a level of $\alpha=1 \%$; $\alpha=5 \%$.
(b) Find the $p$-value of the observation.

Key Concepts

p-Value

The p-value is the probability of obtaining test results at least as extreme as the results observed, under the assumption that the null hypothesis is true. It provides a measure of the evidence against the null hypothesis. A small p-value indicates strong evidence against the null hypothesis, leading to its rejection if the p-value falls below the predetermined level of significance.

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions or inferences about a population based on sample data. It involves setting up two opposing hypotheses – a null hypothesis that represents a baseline or status quo, and an alternative hypothesis that represents the claim being tested. The test then assesses whether the sample evidence is strong enough to reject the null hypothesis in favor of the alternative.

Null and Alternative Hypotheses

The null hypothesis (H?) is a statement of no effect or no difference and is assumed to be true until evidence suggests otherwise. The alternative hypothesis (H? or Ha) is the statement that the data are expected to support, representing a departure from the null. In testing a claim, specifying these hypotheses clearly is crucial, as they define the direction of the statistical test (e.g., one-tailed or two-tailed).

t-Test for Means

A t-test for means is used to determine if there is a significant difference between the sample mean and a hypothesized population mean, especially when the sample size is small and the population standard deviation is unknown. This test relies on the t-distribution, which accounts for variability due to the smaller sample size by incorporating degrees of freedom.

Degrees of Freedom

Degrees of freedom in a t-test refer to the number of independent values that can vary in the analysis while estimating statistical parameters. For a single-sample t-test, the degrees of freedom are calculated as the sample size minus one (n-1). This value is important because it affects the shape of the t-distribution used to determine the critical value and p-value.

Level of Significance

The level of significance, denoted by ?, is a threshold used to determine whether the observed results are statistically significant. Common levels include 1% and 5%. It represents the probability of rejecting the null hypothesis when it is actually true (Type I error). Selecting an appropriate ? level is critical in setting the standards against which the test statistic is compared.

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