Print This Page 83 Confidence Interval For A Population Mean

Print This Page83 Confidence Interval For A Populationmeanfor Point

For point estimation, a single number lies in the forefront even though a standard error is attached. Instead, it is often more desirable to produce an interval of values that is likely to contain the true value of the parameter. Ideally, we would like to be able to collect a sample and then use it to calculate an interval that would definitely contain the true value of the parameter. This goal, however, is not achievable because of sample-to-sample variation. Instead, we insist that before sampling the proposed interval will contain the true value with a specified high probability.

This probability, called the level of confidence, is typically taken as .90, .95, or .99. To develop this concept, we first confine our attention to the construction of a confidence interval for a population mean μ, assuming that the population is normal and the standard deviation σ is known. This restriction helps to simplify the initial presentation of the concept of a confidence interval. Later on, we will treat the more realistic case where σ is also unknown. A probability statement about μ based on the normal distribution provides the cornerstone for the development of a confidence interval.

From Chapter 7, recall that when the population is normal, the distribution of the sample mean is also normal. It has mean μ and standard deviation σ/√n. Here μ is unknown, but σ is a known number because the sample size n is known and we have assumed that σ is known. The normal table shows that the probability is .95 that a normal random variable will lie within 1.96 standard deviations from its mean. For a confidence level of 95%, the interval is constructed as follows:

We then have that the probability is 0.95 that the interval from (sample mean) ± 1.96*(standard error) contains the true population mean μ. This interval can be expressed as:

[\(\bar{x} - 1.96 \frac{\sigma}{\sqrt{n}}\), \(\bar{x} + 1.96 \frac{\sigma}{\sqrt{n}}\)]

This interval, when the population is normal and σ is known, is called a 95% confidence interval for the population mean μ. When the population distribution is normal and σ is known, such intervals can be calculated directly once the sample mean is known.

Paper For Above instruction

Confidence intervals are fundamental tools in statistical inference, providing a range of plausible values for an unknown population parameter based on sample data. When estimating a population mean, the confidence interval offers a probabilistic statement about the location of the true mean, incorporating the variability inherent in sampling. This paper explores the conceptual foundation, construction, and interpretation of confidence intervals for a population mean, emphasizing the classical case where the population is normal with a known standard deviation and extending to scenarios with unknown parameters or larger sample sizes.

At the core of confidence interval construction is the understanding of the sampling distribution of the sample mean. When the population is normally distributed with a known standard deviation, the distribution of the sample mean is also normal, centered at the true mean μ with known standard error, \(\sigma / \sqrt{n}\). The key concept involves leveraging the properties of the standard normal distribution to establish a range that, with a specified confidence level (say 95%), contains the true mean. This leads to the formula for a confidence interval: the sample mean plus or minus a margin of error involving the critical value from the standard normal distribution (1.96 for 95% confidence), the population standard deviation, and the square root of the sample size.

Constructing a confidence interval involves several steps: compute the sample mean, identify the critical z-value corresponding to the desired confidence level, and calculate the margin of error. The resulting interval reflects the sampling variability, not the certainty that any particular observed interval contains the true mean. It is crucial to interpret confidence intervals correctly: they refer to the long-run frequency of such intervals capturing the true parameter if the sampling process is repeated numerous times, rather than the probability that a specific interval from a single experiment contains the parameter.

In practice, the assumption of a known population standard deviation is often unrealistic. When σ is unknown but the sample size is large (typically n ≥ 30), the central limit theorem justifies using the sample standard deviation as an estimator for σ, and the normal approximation remains valid. The confidence interval formula is then adjusted accordingly, replacing σ with the sample standard deviation S and using the z-distribution for large samples.

For smaller samples where the population standard deviation is not known, the t-distribution must be employed. The t-distribution accounts for additional variability due to estimating σ and features wider tails than the normal distribution. The confidence interval is constructed similarly but uses the t-critical value corresponding to the degrees of freedom (n-1) instead of z. This approach ensures approximate coverage probability, especially important when sampling from non-normal populations or with small sample sizes.

The interpretation of confidence intervals ties into the concept of repeated sampling. If numerous independent samples are taken under identical conditions, approximately 95% (or the specified confidence level) of the intervals calculated from those samples will include the true population mean. It is critical to note that once a specific interval is calculated, it either contains the true mean or not—probability no longer applies to that particular interval. The confidence level reflects the long-term performance of the procedure over many repetitions, not the certainty for any single computed interval.

This framework extends beyond the mean to other parameters, such as proportions or differences between groups, requiring analogous calculations and probabilistic interpretations. The central themes involve understanding the diverse sources of variability, proper identification of the critical value, and rigorous interpretation of the resulting intervals in the context of the sampling process.

In conclusion, confidence intervals are invaluable for conveying statistical uncertainty and drawing inference about population parameters. Proper construction, interpretation, and application of confidence intervals require an understanding of the underlying probability models, sampling variability, and the assumptions involved. Whether dealing with normal populations with known standard deviations or applying large-sample approximations with unknown parameters, confidence intervals underpin many statistical analysis and decision-making processes in research and practice.

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