A United Nations report shows the mean family income for...

A United Nations report shows the mean family income for Mexican migrants to the United States is $\$ 27,000$ per year. A FLOC (Farm Labor Organizing Committee) evaluation of 25 Mexican family units reveals the mean to be $\$ 30,000$ with a sample standard deviation of $\$ 10,000 .$ Does this information disagree with the United Nations report? Apply the .01 significance level.

Key Concepts

Hypothesis Testing

Hypothesis testing is the process of making a decision about the properties of a population based on sample data. It involves setting up a null hypothesis, which typically represents a statement of no effect or difference, and an alternative hypothesis that reflects a deviation from that assumption. This framework allows researchers to use sample statistics to infer whether observed differences are likely due to random chance or indicate a true effect.

Significance Level

The significance level, commonly denoted by alpha, is the threshold for determining when a result is statistically significant. It represents the probability of rejecting the null hypothesis when it is actually true (Type I error). A strict significance level, such as 0.01, requires stronger evidence from the sample to reject the null hypothesis, thus reducing the chance of a false positive.

One-Sample t-Test

A one-sample t-test is a statistical method used to compare the mean of a sample to a known or hypothesized population mean, particularly when the population standard deviation is unknown and the sample size is small. This test calculates a t-statistic that quantifies the difference between the sample mean and the population mean relative to the sample’s variability, and then compares this statistic to a t-distribution.

Test Statistic Calculation

The test statistic in a t-test is calculated by subtracting the hypothesized population mean from the sample mean, and then dividing this difference by the estimated standard error, which is derived from the sample standard deviation and sample size. This standardized value is used to determine how far the sample mean is from the population mean in terms of standard errors.

Critical Value and P-Value

The critical value is a cutoff point determined by the chosen significance level and the degrees of freedom, against which the computed test statistic is compared. Alternatively, the p-value, which is the probability of observing a test statistic as extreme as the one calculated given that the null hypothesis is true, can be used. If the p-value is less than the significance level, it indicates sufficient evidence to reject the null hypothesis.

Transcript

Please give Ace some feedback