The Following Statement Is False If A Model For Scores On A
The Following Statement Is Falseif A Model For Scores On A Placeme
The assignment involves multiple statistical concepts, including understanding skewness and the Central Limit Theorem, interpreting confidence intervals correctly, and making hypothesis testing decisions based on significance levels. Specifically, it asks: upon selecting a large random sample of 200 students with a negatively skewed distribution for placement scores, which quantity tends to follow an approximately normal distribution? It also requires correcting a misinterpretation of a confidence interval for third graders' reading scores and determining whether the null hypothesis should be rejected given a specified significance level.
Paper For Above instruction
The question posed about the Central Limit Theorem (CLT) addresses a fundamental concept in statistics: the distribution of sample means. When dealing with a negatively skewed distribution for individual scores, the CLT states that the sampling distribution of the sample mean will tend toward a normal distribution as the sample size increases. Specifically, even if individual data points are not normally distributed, the distribution of the average of a sufficiently large sample will approximate normality. In this case, with n=200 students, the quantity expected to follow an approximately normal distribution is the sample mean placement test score for all freshmen, as the CLT applies to the distribution of the sample mean regardless of the shape of the population distribution, provided the sample size is large enough (Freedman et al., 2007). This supports option (iii): the sample mean placement test score for all freshmen.
Concerning the interpretation of the confidence interval (68, 78), a standard 90% confidence interval implies that if such intervals were constructed repeatedly from many samples, approximately 90% of these intervals would contain the true population mean score. The incorrect interpretation suggests the interval contains the "sample (population) mean," which is a misconception because a confidence interval estimates the population parameter, not the sample statistic or the specific sample mean. Correcting this, the statement should read: "If we were to repeat this study many, many times, we would expect 90% of the resulting intervals to contain the population mean reading achievement score." Therefore, the word "sample" should be crossed out, and "population" should be used as a replacement.
Regarding hypothesis testing, the researcher set a significance level (α) of 5%. Given the confidence interval (68, 78), if the hypothesized mean (H0) is 70 points, since 70 falls within this interval, there is insufficient evidence to reject H0 at the 5% significance level. This is because the confidence interval includes the hypothesized mean, indicating that the data are consistent with H0. Therefore, the decision should be: fail to reject H0.
In summary, the key points are:
1. The quantity with an approximately normal distribution, given a large sample size from a negatively skewed population, is the sample mean (option iii).
2. The incorrect interpretation of the confidence interval should have the word "sample" crossed out and replaced with "population."
3. The decision regarding the null hypothesis, based on the interval containing 70, would be to fail to reject H0.
This comprehensive understanding aligns with standard statistical inference principles and demonstrates the application of the CLT, confidence intervals, and hypothesis testing in practice.
References:
Freedman, D., Pisani, R., & Purves, R. (2007). Statistics (4th ed.). W. W. Norton & Company.
(Note: For a fully detailed paper, additional scholarly references could include texts such as K. Krantz's The Logic of Scientific Discovery or newer publications on statistical inference and hypothesis testing.)