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Chapter 8: Problem 76

Multiple choice: Select the best answer for Exercises 75 to 78. You have an SRS of 23 observations from a Normally distributed population. What critical value would you use to obtain a 98% confidence interval for the mean M of the population if S is unknown? (a) 2.508 (c) 2.326 (e) 2.177 (b) 2.500 (d) 2.183

Short Answer

Expert verified
The critical value is 2.508, option (a).

Step by step solution

01

Determine the t-distribution Characteristics

Since the sample size is 23, which is less than 30, and the population standard deviation is unknown, we use the t-distribution for the confidence interval. We have degrees of freedom (df) equal to the sample size minus 1: \( df = 23 - 1 = 22 \).
02

Identify the Desired Confidence Level

The problem states that we need a 98% confidence interval. This means we want the central 98% of the t-distribution.
03

Calculate the Tail Probabilities

Since the confidence level is 98%, the remaining probability (1 - 0.98 = 0.02) is distributed equally in the two tails of the t-distribution. Each tail will have a probability of \( \frac{0.02}{2} = 0.01 \).
04

Look Up the Critical Value

Using a t-distribution table or calculator, we find the critical value for a 98% confidence interval with 22 degrees of freedom (df). The value corresponding to one tail having a probability of 0.01 and 22 df is approximately 2.508.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

t-distribution
The t-distribution is used in statistics when estimating population parameters when the sample size is small, and the population standard deviation is unknown. It looks similar to the normal distribution but has thicker tails. This makes it a bit more spread out, accommodating for the extra variability that comes from having a smaller sample size. The thick tails allow for greater uncertainty and reflect the higher probability of obtaining values far from the mean. This distribution becomes especially useful when we deal with small samples since using the standard normal distribution would underestimate the amount of variability present.
degrees of freedom
Degrees of freedom are a crucial concept in the calculation of confidence intervals using the t-distribution. They are determined by the formula: the sample size minus one. For example, if you have a sample size of 23, the degrees of freedom would be 22.
  • This one less degree compared to the sample size allows adjustment for the estimation of the population parameter.
  • Degrees of freedom essentially give us an idea of how many independent pieces of information are available to estimate another piece of information.
Every sample used to estimate a parameter loses a degree of freedom for that estimation purpose.
critical value
The critical value is a point on the scale of the t-distribution which corresponds to a specified level of confidence. For example, in the original problem, a 98% confidence level is desired.
To find the critical value:
  • Calculate the tail probability that remains after the desired confidence level. For a 98% confidence interval, this is 0.02 total, or 0.01 in each tail.
  • Use a t-distribution table or calculator, inputting the degrees of freedom (22 in this exercise) to find the t-value that corresponds with 0.01 in each tail.
In this instance, the critical value is 2.508.
sample size
The sample size, denoted usually as 'n', is the number of observations in a sample. In the context of constructing confidence intervals, a larger sample size will generally provide more reliable results due to reduced variability compared to smaller samples.
  • When the sample is less than 30, and the population standard deviation is unknown, the t-distribution is used instead of the normal distribution to account for this variability.
  • The sample size impacts the degrees of freedom, which in turn influences the critical value derived from the t-distribution.
In our exercise, with a sample size of 23, we have to use the t-distribution to calculate confidence intervals.
confidence level
The confidence level represents the degree of certainty that the population parameter lies within the confidence interval. When you set a 98% confidence level, you are saying that if the same population is sampled 100 times, the interval calculated 98 times will include the true population parameter.
  • A higher confidence level means a wider interval, adding more certainty but less precision.
  • A lower confidence level will provide a narrower interval but increases the risk of the interval not containing the population parameter.
Selecting the correct confidence level is a balance between precision and certainty.

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