Paper For Above instruction
The statistical analysis of normal distributions is fundamental in understanding variability in real-world datasets, such as feature film lengths and test scores. This paper explores three key questions concerning normal distributions: calculating probabilities related to film durations, determining top-score thresholds on standardized tests, and elucidating the circumstances under which the Central Limit Theorem (CLT) applies, including the particular case when sample size n equals 1.
Firstly, consider the distribution of film lengths. The problem states that the lengths follow a normal distribution with a mean of 109 minutes and a standard deviation of 12 minutes. The specific question pertains to the proportion of films exceeding 120 minutes, which is two hours. To find this percentage, we convert the raw score to a z-score using the formula:
\[ z = \frac{X - \mu}{\sigma} \]
where \(X = 120\) minutes, \(\mu = 109\), and \(\sigma = 12\). Calculating,
\[ z = \frac{120 - 109}{12} = \frac{11}{12} \approx 0.9167 \]
Referring to standard normal distribution tables or utilizing statistical software, the area to the left of z = 0.9167 is approximately 0.8209. Therefore, the percentage of films shorter than 120 minutes is about 82.09%. Consequently, the percentage of films longer than 120 minutes is:
\[ 100\% - 82.09\% \approx 17.91\% \]
Thus, approximately 17.91% of feature films last longer than two hours.
Secondly, regarding test scores, the distribution is normal with a mean of 73 and a standard deviation of 5. To find the score needed to be among the top 10%, we identify the 90th percentile of the distribution. The
z-score corresponding to the 90th percentile is approximately 1.28, based on z-tables. The raw score corresponding to this z-score is calculated as:
\[ X = \mu + z\sigma = 73 + (1.28)(5) = 73 + 6.4 = 79.4 \]
Therefore, a student would need a score of approximately 79.4 or higher to be among the top 10% of all test takers.
Thirdly, the Central Limit Theorem (CLT) states that the sampling distribution of the sample mean approaches a normal distribution as the sample size \(n\) increases, regardless of the original population distribution, provided that the population has a finite variance. It is essential when applying normal probability calculations to sampling distributions derived from variable data, especially when the data distribution is not normal. The CLT enables approximate normality of the sample mean, facilitating probability calculations.
Importantly, the case when \(n=1\) is explicitly included in the CLT framework, as the theorem considers the distribution of sample means for any sample size \(n\). When \(n=1\), the sample mean is just the individual data point, and its distribution coincides with the original population distribution. In practice, the CLT becomes particularly useful for sample sizes greater than 30, but the theorem technically holds for all finite \(n\), including \(n=1\).
In summary, the CLT is a powerful theorem for approximating the distribution of sample means, aiding in probability calculations in statistics. It applies to all sample sizes, with greater utility as the sample size increases, but does not exclude the case \(n=1\).
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