The cost of a one-carat VS2 clarity, H color diamond from...

The cost of a one-carat VS2 clarity, H color diamond from Diamond Source USA is $\$ 5600$ (Diamond Source website, March 2003 ). A midwestern jeweler makes calls to contacts in the diamond district of New York City to see whether the mean price of diamonds there differs from $\$ 5600$ a. Formulate hypotheses that can be used to determine whether the mean price in New York City differs from $\$ 5600$. b. $A$ sample of 25 New York City contacts provided the prices shown in the file named Diamonds. What is the $p$ -value? c. $A t \alpha=.05,$ can the null hypothesis be rejected? What is your conclusion? d. Repeat the preceding hypothesis test using the critical value approach.

The cost of a one-carat VS2 clarity, H color diamond from Diamond Source USA is $\$ 5600$ (Diamond Source website, March 2003 ). A midwestern jeweler makes calls to contacts in the diamond district of New York City to see whether the mean price of diamonds there differs from $\$ 5600$
a. Formulate hypotheses that can be used to determine whether the mean price in New York City differs from $\$ 5600$.
b. $A$ sample of 25 New York City contacts provided the prices shown in the file named Diamonds. What is the $p$ -value?
c. $A t \alpha=.05,$ can the null hypothesis be rejected? What is your conclusion?
d. Repeat the preceding hypothesis test using the critical value approach.

Key Concepts

Significance Level

The significance level, often denoted by ?, is the threshold for rejecting the null hypothesis. It represents the probability of committing a Type I error, which is rejecting the null hypothesis when it is actually true. Common significance levels are 0.05 or 0.01.

Critical Value Approach

The critical value approach involves determining a threshold value from the t-distribution corresponding to the chosen significance level. If the test statistic falls beyond this value in the direction of the alternative hypothesis, the null hypothesis is rejected. This method provides a clear decision rule based on the calculated t statistic and the critical value.

t-Test

A t-test is a statistical test used to compare the mean of a sample with a known value (or between two groups) when the population standard deviation is unknown and the sample size is small. It assumes that the data are approximately normally distributed and uses the t-distribution to estimate probabilities.

Degrees of Freedom

Degrees of freedom refer to the number of independent pieces of information used to estimate a parameter. In the context of a one-sample t-test, the degrees of freedom typically equal the sample size minus one. They affect the shape of the t-distribution, which in turn impacts the critical values used in the hypothesis test.

Alternative Hypothesis

The alternative hypothesis is the statement that contradicts the null hypothesis. It posits that there is a difference or effect, such as the population mean differing from a known value. This hypothesis is what researchers aim to support and is usually directional or non-directional depending on the context of the test.

Null Hypothesis

The null hypothesis is a default statement asserting that there is no effect or difference. In hypothesis testing, it is the assumption that the population parameter (such as the mean) equals a specified value. It serves as a starting point for statistical analysis and testing, and the evidence from a sample is used to determine whether this assumption is reasonable.

p-value

The p-value is the probability of obtaining a test statistic at least as extreme as the one observed, assuming that the null hypothesis is true. It quantifies the strength of evidence against the null hypothesis; a low p-value indicates that the observed data is unlikely under the null hypothesis, leading to its rejection.

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