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Chapter 12: Problem 28

The formula used to calculate a confidence interval for the mean of a normal population is $$ \bar{x} \pm(t \text { critical value }) \frac{s}{\sqrt{n}} $$ What is the appropriate \(t\) critical value for each of the following confidence levels and sample sizes? a. \(90 \%\) confidence, \(n=12\) b. \(90 \%\) confidence, \(n=25\) c. \(95 \%\) confidence, \(n=10\)

Short Answer

Expert verified
The appropriate t critical values for the given cases are: a. 90% confidence, n=12: \(t = 1.796\) b. 90% confidence, n=25: \(t = 1.711\) c. 95% confidence, n=10: \(t = 2.262\)

Step by step solution

01

Determine the degrees of freedom for each case

For each of the given cases, we need to compute the degrees of freedom (df) using the formula df = n - 1. a. 90% confidence, n=12 df = 12 - 1 = 11 b. 90% confidence, n=25 df = 25 - 1 = 24 c. 95% confidence, n=10 df = 10 - 1 = 9
02

Consult the t-distribution table for critical values

Now we will look up the t critical values in the t-distribution table for each case, given the confidence level and degrees of freedom calculated in step 1. a. 90% confidence, df = 11 The t critical value for 90% confidence with 11 degrees of freedom is 1.796. b. 90% confidence, df = 24 The t critical value for 90% confidence with 24 degrees of freedom is 1.711. c. 95% confidence, df = 9 The t critical value for 95% confidence with 9 degrees of freedom is 2.262.
03

Report the t critical values

Now we have found the t critical values for each of the given cases: a. 90% confidence, n=12: The appropriate t critical value is 1.796. b. 90% confidence, n=25: The appropriate t critical value is 1.711. c. 95% confidence, n=10: The appropriate t critical value is 2.262.

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Key Concepts

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

t-distribution
In statistics, the t-distribution is crucial for estimating population parameters when the sample size is small or the population variance is unknown. It resembles the normal distribution but has thicker tails. These thicker tails indicate that there is a greater probability of observing values far from the mean, which is especially important when dealing with small samples.
When calculating confidence intervals, the t-distribution is used instead of the normal distribution if the sample size is below 30 or the population standard deviation is not known. This distribution helps in determining how much the sample mean may vary from the population mean.
  • The t-distribution becomes more like the normal distribution as the sample size increases.
  • The shape of the t-distribution depends on the degrees of freedom.
degrees of freedom
Degrees of freedom (df) represent the number of values in a calculation that are free to vary. This concept is essential when conducting statistical analyses like the t-test.
For confidence intervals of the mean, degrees of freedom are calculated as the number of observations minus one, written as \( df = n - 1 \), where \( n \) is the sample size. Degrees of freedom play a pivotal role in determining the shape of the t-distribution.
A greater number of degrees of freedom results in a distribution that closely resembles the standard normal distribution. Conversely, a smaller number reflects broader tails, meaning more variability.
  • Helps adjust the influence of sample size in statistical tests.
  • Vital for obtaining accurate critical values from the t-distribution table.
critical value
The critical value is a crucial component when constructing confidence intervals. It indicates the limit at which you still consider the sample mean to be representative of the population mean.
When using the t-distribution for smaller sample sizes, you need to look up these critical values based on confidence levels and degrees of freedom in the t-distribution table.
Critical values tell us how much we can stretch the interval around the sample mean. For instance:
  • A 90% confidence level implies that you expect the calculation to be accurate 90% of the time.
  • The higher the confidence level, the larger the critical value, resulting in a wider interval.

This concept ensures that the interval has a specific probability of containing the true population mean.
sample size
Sample size, denoted as \( n \), is the number of observations or data points collected in a survey or experiment. It is fundamental to the calculation of means and variances.
A larger sample size typically results in a more reliable estimate of the population mean, as it provides more information and reduces the margin of error. Sample size directly impacts the degrees of freedom, which in turn affects the choice of the t-distribution over the normal distribution.
  • Increasing sample size decreases the width of the confidence interval.
  • Helps achieve more precise and accurate estimates.

Understanding the relationship between sample size and the stability of your statistical estimates is critical in ensuring the reliability and validity of your findings.

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Most popular questions from this chapter

USA TODAY (October 14,2016 ) reported that Americans spend 4.1 hours per weekday checking work e-mail. This was an estimate based on a survey of 1004 white-collar workers in the United States. a. Suppose that you would like to know if there is evidence that the mean time spent checking work e-mail for white-collar workers in the United States is more than half of the 8 -hour work day. What would you need to assume about the sample in order to use the given sample data to answer this question? b. Given that any concerns about the sample were satisfactorily addressed, carry out a test to decide if there is evidence that the mean time spent checking work e-mail for white-collar workers in the United States is more than half of the 8 -hour work day. Suppose that the sample standard deviation was \(s=1.3\) hours.

The paper "The Effects of Adolescent Volunteer Activities on the Perception of Local Society and Community Spirit Mediated by Self-Conception" (Advanced Science and Technology Letters [2016]: 19-23) describes a survey of a large representative sample of middle school children in South Korea. One question in the survey asked how much time per year the children spent in volunteer activities. The sample mean was 14.76 hours and the sample standard deviation was 16.54 hours. a. Based on the reported sample mean and sample standard deviation, explain why it is not reasonable to think that the distribution of volunteer times for the population of South Korean middle school students is approximately normal. b. The sample size was not given in the paper, but the sample size was described as "large." Suppose that the sample size was 500 . Explain why it is reasonable to use a one-sample \(t\) confidence interval to estimate the population mean even though the population distribution is not approximately normal. c. Calculate and interpret a confidence interval for the mean number of hours spent in volunteer activities per year for South Korean middle school children. (Hint: See Example 12.7.)

An automobile manufacturer decides to carry out a fuel efficiency test to determine if it can advertise that one of its models achieves \(30 \mathrm{mpg}\) (miles per gallon). Six people each drive a car from Phoenix to Los Angeles. The resulting fuel efficiencies (in miles per gallon) are: \(\begin{array}{ll}30.3 & 29.6\end{array}\) \(\begin{array}{llll}27.2 & 29.3 & 31.2 & 28.4\end{array}\) Assuming that fuel efficiency is normally distributed, do these data provide evidence against the claim that actual mean fuel efficiency for this model is (at least) \(30 \mathrm{mpg}\) ?

The Economist collects data each year on the price of a Big Mac in various countries around the world. A sample of McDonald's restaurants in Europe in July 2016 resulted in the following Big Mac prices (after conversion to U.S. dollars): \(\begin{array}{llll}4.44 & 3.15 & 2.42 & 3.96\end{array}\) \(\begin{array}{llll}4.51 & 4.17 & 3.69 & 4.62\end{array}\) \(\begin{array}{lll}3.80 & 3.36 & 3.85\end{array}\) The mean price of a Big Mac in the U.S. in July 2016 was \$5.04. For purposes of this exercise, you can assume it is reasonable to regard the sample as representative of European McDonald's restaurants. Does the sample provide convincing evidence that the mean July 2016 price of a Big Mac in Europe is less than the reported U.S. price? Test the relevant hypotheses using \(\alpha=0.05 .\) (Hint: See Example 12.12.)

Explain the difference between \(\bar{x}\) and \(\mu_{\vec{x}}\).
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