Assume that a simple random sample has been selected from a...

Assume that a simple random sample has been selected from a normally distributed population and test the given claim. Unless specified by your instructor, use either the traditional method or P-value method for testing hypotheses. Identify the mull and alternative hypotheses, test statistic, $P$-value (or range of P-values), critical value(s), and state the final conclusion that addresses the original claim. A simple random sample of 25 filtered $100 \mathrm{mm}$ cigarettes is obtained, and the tar content of each cigarette is measured. The sample has a mean of $13.2 \mathrm{mg}$ and a standard deviation of $3.7 \mathrm{mg}$ (based on Data Set 4 in Appendix $\mathrm{B}$ ). Use 20.05 significance level to test the chim that the mean tar content of filtered $100 \mathrm{mm}$ cigarettes is loss than $21.1 \mathrm{mg}$ which is the mean for unfiltered king sixe cigarettes. What do the results suggest about the effectiveness of the filters?

Assume that a simple random sample has been selected from a normally distributed population and test the given claim. Unless specified by your instructor, use either the traditional method or P-value method for testing hypotheses. Identify the mull and alternative hypotheses, test statistic, $P$-value (or range of P-values), critical value(s), and state the final conclusion that addresses the original claim.
A simple random sample of 25 filtered $100 \mathrm{mm}$ cigarettes is obtained, and the tar content of each cigarette is measured. The sample has a mean of $13.2 \mathrm{mg}$ and a standard deviation of $3.7 \mathrm{mg}$ (based on Data Set 4 in Appendix $\mathrm{B}$ ). Use 20.05 significance level to test the chim that the mean tar content of filtered $100 \mathrm{mm}$ cigarettes is loss than $21.1 \mathrm{mg}$ which is the mean for unfiltered king sixe cigarettes. What do the results suggest about the effectiveness of the filters?

Key Concepts

Conclusion in Hypothesis Testing

In hypothesis testing, once the test statistic has been calculated and compared to the critical value or the p-value has been obtained, a final conclusion is drawn. This conclusion either rejects or fails to reject the null hypothesis, thus indicating whether there is sufficient statistical evidence to support the claim expressed by the alternative hypothesis.

Significance Level

The significance level, often denoted by alpha, is the pre-determined probability of committing a Type I error. It sets the standard by which the p-value or test statistic is judged. A result is generally considered statistically significant if the p-value is less than the significance level, leading to the rejection of the null hypothesis.

Critical Value

A critical value is a threshold derived from the chosen significance level and the relevant probability distribution. If the test statistic falls in the region beyond this threshold, the null hypothesis is rejected. The selection of the critical value helps control the probability of a Type I error, which is the error of rejecting a true null hypothesis.

p-Value

The p-value represents the probability of obtaining a test statistic at least as extreme as the one calculated from the sample data, assuming the null hypothesis is true. It provides a measure of the strength of the evidence against the null hypothesis, with lower p-values indicating stronger evidence in favor of the alternative hypothesis.

Null and Alternative Hypotheses

The null hypothesis (H0) represents the assumption that there is no effect or difference, typically stating the population parameter equals a specific value. The alternative hypothesis (H1) is formulated to reflect the claim or effect the researcher is interested in testing; it can be one-sided or two-sided. Together, these hypotheses establish the framework for the decision-making process in hypothesis testing.

Test Statistic

The test statistic is a numerical summary computed from sample data that, under the null hypothesis, follows a known probability distribution. It is used to determine how far the sample statistic diverges from the hypothesized parameter, effectively quantifying the evidence against the null hypothesis.

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about a population parameter based on sample data. It involves setting up two competing hypotheses—the null hypothesis representing a default or status quo scenario, and the alternative hypothesis representing a claim to be tested. The process encompasses calculating a test statistic, comparing it to a critical value, and making a decision about the null hypothesis.

t-Distribution

When the sample size is small and the population standard deviation is unknown, as is common in many practical situations, the t-distribution is used instead of the normal distribution. The t-distribution accounts for additional uncertainty due to estimating the population standard deviation from the sample, and its shape depends on the degrees of freedom.

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