A car dealer recommends that transmissions be serviced at...

A car dealer recommends that transmissions be serviced at 30,000 miles. To see whether her customers are adhering to this recommendation, the dealer selects a random sample of 40 customers and finds that the average mileage of the automobiles serviced is $30,456 .$ The standard deviation of the population is 1684 miles. By finding the $P$ -value, determine whether the owners are having their transmissions serviced at 30,000 miles. Use $\alpha=0.10$. Do you think the $\alpha$ value of 0.10 is an appropriate significance level?

Key Concepts

P-value

The P-value is the probability of obtaining test results at least as extreme as the observed results, assuming that the null hypothesis is true. It helps determine the significance of the results: a small P-value indicates that the observed effect is unlikely to have occurred by chance, thus providing evidence to reject the null hypothesis in favor of the alternative hypothesis.

Significance Level

The significance level (denoted as ?) is the threshold used to decide whether the P-value indicates a statistically significant result. It represents the probability of rejecting the null hypothesis when it is actually true, also known as the Type I error rate. A higher significance level, such as 0.10, makes it easier to reject the null hypothesis, but it also increases the risk of a false positive.

Type I Error

A Type I error occurs when the null hypothesis is wrongly rejected, meaning that a difference is detected when none actually exists. The significance level (?) directly controls the likelihood of making a Type I error. Understanding and managing the risk of Type I error is crucial when choosing an appropriate significance level for a statistical test.

Z-Test for Means

The Z-test for means is a statistical test used to determine whether there is a significant difference between the sample mean and the population mean when the population standard deviation is known. It uses the Z-statistic, which measures how many standard errors the sample mean is away from the hypothesized population mean. This test is appropriate when the sample size is large or when the population distribution is normal.

Null and Alternative Hypotheses

The formulation of null and alternative hypotheses is central to the hypothesis testing process. The null hypothesis (H0) typically represents the status quo or a statement of no effect or no difference, while the alternative hypothesis (Ha) represents a claim that there is an effect or a difference. Choosing the right hypotheses guides the direction and type of test to be performed.

Hypothesis Testing

Hypothesis testing is a statistical method used to decide whether there is enough evidence in a sample to infer that a certain condition holds for the entire population. It involves formulating a null hypothesis, which represents a default position or claim to be tested, and an alternative hypothesis, which represents a competing claim. Based on the sample data, a test statistic is calculated to decide whether to reject the null hypothesis in favor of the alternative.

Sampling Distribution and the Central Limit Theorem

The sampling distribution of the sample mean describes how the sample means would vary if many samples were taken from the same population. The Central Limit Theorem states that, for a sufficiently large sample size, the sampling distribution of the mean will be approximately normally distributed, regardless of the underlying population distribution. This principle justifies the use of normal-based methods like the Z-test when the sample size is large.

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