Homes in a nearby college town have a mean value of...

Homes in a nearby college town have a mean value of $88,950$dollar . It is assumed that homes in the vicinity of the college have a higher mean value. To test this theory, a random sample of 12 homes is chosen from the college area. Their mean valuation is $92,460$dollar, and the standard deviation is $5200$dollar. Complete a hypothesis test using $\alpha=0.05 .$ Assume prices are normally distributed. a. Solve using the $p$ -value approach. b. Solve using the classical approach.

Homes in a nearby college town have a mean value of $88,950$dollar . It is assumed that homes in the vicinity of the college have a higher mean value. To test this theory, a random sample of 12 homes is chosen from the college area. Their mean valuation is $92,460$dollar, and the standard deviation is $5200$dollar. Complete a hypothesis test using $\alpha=0.05 .$ Assume prices are normally distributed.
a. Solve using the $p$ -value approach.
b. Solve using the classical approach.

Key Concepts

Significance Level

The significance level, often denoted by alpha (?), is the threshold for making a decision in hypothesis testing. It represents the probability of rejecting the null hypothesis when it is, in fact, true (a type I error). Commonly set at 0.05, this level is used to determine whether the p-value or test statistic is extreme enough to reject the null hypothesis.

p-Value Approach

In the p-value approach, after computing the test statistic using the sample data, the p-value is determined by finding the probability that the test statistic would be at least as extreme as the one observed, assuming that the null hypothesis is true. This probability is then compared to the significance level, and if the p-value is less than the significance level, the null hypothesis is rejected.

Classical Approach

The classical approach, also known as the critical region method, involves comparing the calculated test statistic to a critical value derived from the appropriate statistical distribution (in this case the t-distribution) at a specific significance level. If the test statistic falls into the critical region, the null hypothesis is rejected. This method uses tables or computed critical values to define rejection regions based on degrees of freedom and the specified significance level.

Hypothesis Testing

This is a statistical method used to decide whether there is enough evidence in a sample to conclude that a certain condition is true for the entire population. It involves setting up a null hypothesis (typically representing no effect or status quo) and an alternative hypothesis (representing the effect or difference of interest) and then making a decision based on the sample data and a pre-determined significance level.

One-Sample t-Test

The one-sample t-test is a procedure used to determine whether the mean of a single sample differs significantly from a known or hypothesized population mean. It is particularly applicable when the sample size is small and the population standard deviation is unknown, requiring the use of the t-distribution with degrees of freedom equal to the sample size minus one.

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