Question

Let a random sample of size 17 from the normal distribution \(N\left(\mu, \sigma^{2}\right)\) yield \(\bar{x}=4.7\) and \(s^{2}=5.76 .\) Determine a 90 percent confidence interval for \(\mu .\)

Short answer

The 90 percent confidence interval for \(\mu\) is between 3.77 and 5.63.

Step-by-step solution

Identification of the Given Parameters

Identify the given parameters from the problem. The sample mean is \(\bar{x} = 4.7\), the sample variance is \(s^{2} = 5.76\), and the sample size is \(n = 17\).

Finding the Z-Score

The confidence level is 90 percent, so this leaves 10 percent in the two tails of the normal distribution. Because the tails are on both sides of the distribution, divide the remaining area by 2, resulting in 5 percent (0.05) in each tail. Look up the Z-score that corresponds to an area of 0.95 in a standard normal table or use a Z-score calculator. The Z-score is approximately \(Z_{0.05} = 1.645\).

Plug into the Formula of Confidence Interval

Plugging these values into the confidence interval formula \(\bar{x}\pm Z_{\frac{\alpha}{2}}\sqrt{\frac{s^{2}}{n}}\) yields \(4.7 \pm 1.645 \times \sqrt{5.76/17} = 4.7 \pm 1.645 \times 0.57\). The result is \(4.7 \pm 0.93\).

Calculate the Range

Subtract and add the resulting value to the sample mean to get the confidence interval. The confidence interval is \(4.7-0.93 = 3.77\) to \(4.7+0.93 = 5.63\).

Key concepts

Normal Distribution

The normal distribution is a continuous probability distribution that's symmetrical around its mean, often referred to as a bell curve because of its shape. It's characterized by two main parameters:

• Mean (\( \mu \)
• Variance (\( \sigma^2 \)

These parameters determine the location and the spread of the distribution. The mean is the central point, and the variance measures how spread out the values are around the mean. Many natural phenomena follow a normal distribution, making it a fundamental concept in statistics and probability. In the context of confidence intervals, the normal distribution helps in determining how sample data can be used to estimate a population parameter.

Sample Variance

Sample variance is a measure of the variability or spread of a set of data points in a sample. It's denoted by \( s^2 \). The formula for calculating sample variance is:\[ s^2 = \frac{1}{n-1} \sum{(x_i - \bar{x})^2} \]where

• \( x_i \) is each individual data point,
• \( \bar{x} \) is the sample mean,
• \( n \) is the sample size.

The sample variance provides an estimate of the population variance and is essential in constructing confidence intervals. By knowing the spread of the data, you can better understand the amount of variation present, which feeds into the reliability of your estimates.

Z-Score

A Z-score, or standard score, indicates how many standard deviations a data point is from the mean. It's calculated using the formula:\[ Z = \frac{(X - \mu)}{\sigma} \]For confidence intervals, Z-scores corresponding to specific confidence levels (such as 90%, 95%) are critical. These scores are derived from the standard normal distribution:

• A 90% confidence interval uses a Z-score of approximately 1.645.
• A larger Z-score would mean more confidence but a wider interval.

Z-scores enable comparison across different data sets and facilitate the construction of confidence intervals by determining the critical value that's multiplied by the standard error.

Random Sample

A random sample is a subset of individuals chosen from a larger population in such a way that every possible sample of the same size has an equal probability of being selected. Random sampling minimizes biases and ensures the generalizability of the results to the whole population. Key points include:

• Improves the reliability of the statistical analyzes.
• Allows for estimation of population parameters with a known level of accuracy.
• Ensures that the findings are not skewed by any particular trend in the sample data.

In the exercise, the random sample from a normal distribution helps in making accurate confidence interval estimates about the mean.