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Evaluate the following statements related to statistical tests and their assumptions, interpretations, and applications:

1. In an experiment involving matched pairs, a sample of 14 pairs of observations is collected. The degree of freedom for the t statistic is 13.

2. When testing the difference between two means from two independent populations, equal sample sizes are necessary to use the Z statistic.

3. The sampling distribution of the difference in means approximates normality when neither population is normal, provided the total combined sample size is at least 30.

4. If the confidence interval for the difference between two population means is from 0.5 to 2.5 at a 95% confidence level, we can conclude there is a statistically significant difference between the means.

5. When comparing two independent population means with unknown but equal variances, a pooled variance estimate is used if the population standard deviations are not known.

6. The chi-square distribution is skewed to the left.

7. In a contingency table, when all expected frequencies equal observed frequencies, the chi-square statistic equals 1.

8. Equal expected and observed frequencies in a contingency table imply perfect dependence between rows and columns.

9. As the difference between observed and expected frequencies decreases in a chi-square test, the probability of concluding independence increases.

10. In a chi-square independence test, the expected frequencies are based on the alternative hypothesis.

11. When rejecting a null hypothesis at a significance level of 0.01, it is always rejected at 0.05.

12. If a null hypothesis is rejected at 0.05 significance level, it is always rejected at 0.01 significance level.

13. In testing whether at least 40% of residences in a California county are insured against earthquakes, a sample of 337 reveals 133 insured homes. Calculate the appropriate test statistic for the hypothesis that the true proportion is at least 0.40.

14. Using a Z statistic instead of a t statistic when comparing two means from small samples affects the probability of not rejecting the null hypothesis.

15. A fast-food restaurant claims that the average waiting time has decreased from 7.5 minutes. A sample of 25 customers has a mean wait of 6.64 minutes and a standard deviation of 2 minutes. Test this claim at 5% and 1% significance levels.

16. A study shows that 33% of freshmen received at least one A. In a recent sample of 500 students, 180 had at least one A. Test whether grade inflation has increased this percentage at 5% and 1% significance levels.

17. A manufacturing firm wants to determine if the average completion time has increased from 6 days. A sample of 36 jobs has a mean of 6.32 days and a standard deviation of 1.2 days. Test whether the average time has increased at 5% and 10% significance levels.

18. Comparing profit growth proportions between industry sectors involves testing if the proportion of companies with growth in one sector exceeds that in another sector, given specific sample data.

19. A coach claims that a six-month training program significantly reduces the time to run 1500 meters. Times are recorded before and after training for five runners. Test for a significant decrease at 0.05 and 0.01 significance levels.

20. Testing whether the mean time for a certain activity differs between two groups involves setting hypotheses, computing the test statistic, and finding critical values. Use data provided for two groups with known means, standard deviations, and sample sizes, assuming equal variances.

21. Grade distributions in a statistics course are compared to historical proportions using a chi-square goodness-of-fit test, to see if the current course's grade distribution differs significantly from the past, at 5% and 1% significance levels.

22. The number of hospital admissions over a two-year period is examined for a sample of 300 city residents and compared to national percentages. The goal is to assess if the city's health profile differs significantly from national data at 5% and 1% significance levels.

Paper For Above instruction

In statistical hypothesis testing, understanding the assumptions, applications, and interpretations of various tests is essential for accurate analysis. The statements and questions provided span a broad array of concepts, including matched pairs tests, independent sample comparisons, chi-square tests, confidence intervals, and proportion tests, each pivotal in conducting rigorous statistical analysis.

The first statement pertains to the degrees of freedom in a matched pairs t-test, which is computed as n - 1, where n is the number of pairs. For 14 pairs, degrees of freedom are 13. This is correct, as the matched pairs method inherently reduces the data to differences, and the degrees of freedom depend on the number of pairs, not individual observations.

Next, the necessity of equal sample sizes to employ the Z statistic when testing the difference between two independent means is false. The Z test for two independent means assumes known population variances, not necessarily equal sample sizes, although large samples and equal variances simplify computation. In practice, unequal sample sizes are common and manageable, especially with variance adjustments or Welch's t-test.

The third statement correctly points out the importance of sample size in approximating normality of the sampling distribution of the difference in means when neither population is normal. By the Central Limit Theorem, the sum of independent samples becomes approximately normal if the total sample size exceeds 30.

Regarding the confidence interval from 0.5 to 2.5, the conclusion that there is a significant difference hinges on whether this interval contains zero. Since it does not, the interval suggests the difference is statistically significant at the 95% confidence level, allowing us to reject the null hypothesis of no difference.

The fifth statement discusses the pooled variance estimate, used in t-tests for independent samples when the population variances are assumed equal but unknown. This pooled estimate combines information from both samples, improving the estimate's accuracy under the assumption of equal variances.

The chi-square distribution is right-skewed (not left), especially for small degrees of freedom, which is a key feature of its shape. The statement claiming skewness to the left is false. In contingency tables, a chi-square statistic equals 1 when observed and expected frequencies are identical, but this is a specific case and not a general rule.

When the observed frequencies match the expected frequencies exactly, the chi-square statistic is zero, indicating perfect fit, not necessarily dependence. Dependence or independence in contingency tables is assessed statistically, not through perfect observed-expected matches alone.

The interpretation that decreasing differences between observed and expected frequencies increases the probability of concluding dependence is reversed; smaller differences tend to support independence. Larger deviations increase the likelihood of rejecting independence.

In chi-square tests, the expected frequencies are derived under the null hypothesis of independence, not the alternative.

Regarding significance levels, rejecting a null hypothesis at 0.01 indicates strong evidence; such hypotheses are also rejected at 0.05 levels, so they are not always or never rejected, just generally at the higher levels. Rejections at 0.05 typically lead to rejection at 0.01, but not vice versa, reflecting the stricter criterion.

In proportions testing, the sample proportion of insured homes (133/337 ≈ 0.395) compares to the hypothesized proportion of 0.40. Computing the z-statistic allows testing whether the true proportion is at least 0.40, with the result guiding conclusions about insurance prevalence.

When using the z test instead of a t test with small samples, the probability of not rejecting the null hypothesis can be affected, often decreasing because z tests do not account for small sample variability as effectively. Thus, the probability of Type II error may increase or decrease depending on the context.

In testing claims of decreased waiting times or increased grades, recent samples provide data to perform z or t-tests, comparing sample means to claimed population means, considering standard deviations and sample sizes, and deriving test statistics. Critical values determine whether observed differences are statistically significant at specified levels.

Similarly, comparisons between gender or sector-based proportions involve hypothesis tests—either for proportions (using z) or means—assessing if the observed data supports the presence of a true difference, or if differences could be due to sampling variation.

In a paired sample scenario—such as measuring times before and after training—the paired t-test evaluates whether the mean difference is significantly different from zero, indicating effectiveness. The hypotheses are set accordingly, and the t-statistic computed from the differences determines significance at specified alpha levels.

For tests comparing two independent means under the assumption of equal variances, the formula includes pooled variance; hypotheses specify whether the mean of group 1 exceeds that of group 2; critical values are obtained from the t-distribution with appropriate degrees of freedom. The conclusion depends on whether the test statistic exceeds the critical value for a one-tailed test.

Chi-square goodness-of-fit tests compare observed to expected frequencies to assess if the sample distribution differs from expected proportions. The null hypotheses specify no difference; the test statistic is derived, and significance levels determine whether such differences are statistically meaningful.

Similarly, chi-square tests for independence assess whether variables are related beyond chance, based on contingency table data. Small deviations support independence; large deviations lead to rejection of independence, indicating an association between variables like hospital admissions or demographic categories.

In summary, the statements encompass the critical concepts in statistical inference, from hypothesis testing to estimation, emphasizing the importance of assumptions, sample size, and correct interpretation. Proper application of these principles ensures valid conclusions across various contexts involving means, proportions, and categorical data.

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