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Chapter 21: Problem 9

In this chapter technical details were presented for finding a \(t\) -multiplier for constructing a confidence interval for a population mean. Use appropriate software, a calculator, or a website to find the \(t\) -multiplier for the following situations. a. \(95 \%\) confidence interval, \(n=200\) b. \(95 \%\) confidence interval, \(n=50\) c. \(90 \%\) confidence interval, \(n=200\) d. \(90 \%\) confidence interval, \(n=50\)

Short Answer

Expert verified
For a 95% interval: t-multipliers are 1.96 (n=200) and 2.009 (n=50). For a 90% interval: t-multipliers are 1.645 (n=200) and 1.676 (n=50).

Step by step solution

01

Understand the Concept of the t-Multiplier

The t-multiplier is a value from the t-distribution that helps construct confidence intervals for a population mean when the sample size is small or the population standard deviation is unknown. The t-distribution is similar to the standard normal distribution but accounts for additional variability due to smaller sample sizes.
02

Calculate Degrees of Freedom

For each scenario, calculate the degrees of freedom ( df = n - 1 ). This is essential as the t-distribution changes shape based on this value. Compute this for each given sample size.
03

Step 3a: 95% Confidence Interval for n = 200

Calculate degrees of freedom: df = 200 - 1 = 199 . Find the t-multiplier for a 95% confidence interval using statistical software or a t-distribution table, which for this large sample size, closely approximates 1.96.
04

Step 3b: 95% Confidence Interval for n = 50

Calculate degrees of freedom: df = 50 - 1 = 49 . Find the t-multiplier from a t-distribution table or calculator for 95% confidence, which is approximately 2.009.
05

Step 3c: 90% Confidence Interval for n = 200

Calculate degrees of freedom: df = 200 - 1 = 199 . From a t-distribution table, find the t-multiplier for a 90% confidence interval, which is approximately 1.645.
06

Step 3d: 90% Confidence Interval for n = 50

Calculate degrees of freedom: df = 50 - 1 = 49 . Find the t-multiplier from the t-distribution table for a 90% confidence interval, approximately 1.676.

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Key Concepts

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

Confidence Interval
A confidence interval is a range of values that aims to estimate an unknown population parameter, like the mean. Imagine you took multiple samples from a population and calculated the mean from each. A confidence interval tells you how sure you can be that a population mean falls within a certain range. The width of this interval depends on factors such as sample size and the confidence level chosen.
  • Confidence Level: The percentage of all possible samples that can be expected to include the population parameter. For example, a 95% confidence level suggests that if you were to take 100 different samples and compute a confidence interval for each, roughly 95 of those intervals would contain the true population mean.
  • Sample Size: Larger samples tend to produce more accurate (narrower) confidence intervals, as they provide more information about the population.
To summarize, a confidence interval provides a probable range of values for the population mean, with associated confidence.
Degrees of Freedom
Degrees of freedom (df) describe the number of values in a calculation that can vary independently. In relation to the t-distribution, degrees of freedom are crucial as they affect the shape of the t-distribution.
  • Calculation: Usually calculated as the sample size minus one (df = n - 1). This account for the loss of a degree of freedom once a sample mean is calculated.
  • Effect: As degrees of freedom increase, the t-distribution approaches the normal distribution. When the sample size is smaller, the t-distribution is wider and has heavier tails.
Degrees of freedom allow us to use the t-distribution to make inferences about the population mean, especially when the population standard deviation is unknown or the sample size is small.
T-Multiplier
The t-multiplier is a critical value derived from the t-distribution, used to construct confidence intervals when dealing with small samples or an unknown population standard deviation. Its value is determined by the confidence level chosen and the degrees of freedom.
  • Selection: Identified using statistical software or a t-distribution table for the specific confidence level and degrees of freedom in your study.
  • Purpose: Multiplied by the standard error to widen the range of your confidence interval appropriately.
By providing the correct t-multiplier, one ensures that the calculated confidence interval will correctly reflect the needed confidence level, accounting for sample size and variability.
Population Mean
The population mean is the average of a given characteristic across an entire population. While gathering data for a whole population is often infeasible, sample means can be used to estimate it.
  • Sample Mean: Calculated as the sum of all sampled observations divided by the number of observations. It serves as an estimate of the population mean.
  • Purpose: Knowing the population mean helps researchers and statisticians understand trends and make informed decisions.
Utilizing sample data, the goal is to make predictions or inferences about the population mean, ensuring that the sample accurately reflects the population as a whole.

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Most popular questions from this chapter

Refer to Exercises 18 and \(19 .\) The 200 men in the sample had a mean height of 68.2 inches, with a standard deviation of 2.7 inches. The 200 women had a mean height of 63.1 inches, with a standard deviation of 2.5 inches. Assuming these were independent samples, compute an approximate \(95 \%\) confidence interval for the mean difference in heights between British males and females. Interpret the resulting interval in words that a statistically naive reader would understand.

Parts (a) through (d) below provide additional results for Case Study 21.1 . For each of the parts, compute an approximate \(95 \%\) confidence interval for the difference in mean symptom scores between the placebo and calcium-treated conditions for the symptom listed. In each case, the results given are mean \(\pm\) standard deviation. There were 228 participants in the placebo group and 212 in the calcium-treated group. a. Mood swings: placebo \(=0.70 \pm 0.75 ;\) calcium \(=0.50 \pm 0.58\) b. Crying spells: placebo \(=0.37 \pm 0.57 ;\) calcium \(=0.23 \pm 0.40\) c. Aches and pains: placebo \(=0.49 \pm 0.60 ;\) calcium \(=0.31 \pm 0.49\) d. Craving sweets or salts: placebo \(=0.60 \pm 0.78 ;\) calcium \(=0.43 \pm 0.64\)

In this chapter technical details were presented for finding a \(t\) -multiplier for constructing a confidence interval for a population mean. Use appropriate software, a calculator, or a website to find a confidence interval for the population mean for the following situations. Use the \(t\) -multiplier in each case. a. Refer to Table 21.2. Find a 95\% confidence interval for the population mean age of Australian males in couples who separated during the time period of the study by Butterworth et al (2008). Relevant numbers from the table are the sample mean of 37.65 years, SEM of 0.75, and sample size of 114 b. Repeat part (a) for the population of males in couples who remained intact. Relevant numbers from the table are the sample mean of 39.18 years, SEM of \(0.19,\) and sample size of 1384 c. Repeat part (a) using a confidence level of \(90 \%\)

Notice that in Table 14 (on page 51 of Original Source 5 ) some of the intervals are much wider than others. For instance, the \(95 \%\) confidence interval for the percent of time with hands off the wheel while not reading or writing is from 0.97 to \(1.93,\) compared with the interval from 4.24 to 34.39 while reading or writing. What two features of the data do you think are responsible for some intervals being wider than others? (Hint: What two features of the data determine the width of a 95\% confidence interval for a mean? The answer is the same in this situation.)

Using the data presented by Hand and colleagues (1994) and discussed in previous chapters, we would like to estimate the average age difference between husbands and wives in Britain. Recall that the data consisted of a random sample of 200 couples. Following are two methods that were used to construct a confidence interval for the difference in ages. Your job is to figure out which method is correct: Method 1: Take the difference between the husband's age and the wife's age for each couple, and use the differences to construct an approximate \(95 \%\) confidence interval for a single mean. The result was an interval from 1.6 to 2.9 years. Method 2: Use the method presented in this chapter for constructing an approximate confidence interval for the difference in two means for two independent samples. The result was an interval from -0.4 to 4.3 years. Explain which method is correct, and why. Then interpret the confidence interval that resulted from the correct method.
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