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Ch. 8 - Hypothesis Testing with Two Samples
Larson - Elementary Statistics: Picturing the World 8th Edition
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoLarson 8th EditionCh. 8 - Hypothesis Testing with Two SamplesProblema 8.CR.6
Capitolo 8, Problema 8.CR.6

In Exercises 3–6, construct the indicated confidence interval for the population mean . Which distribution did you use to create the confidence interval?


c=0.90, x̅=8.21, σ=0.62, n=8

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Step 1: Identify the given values in the problem. The confidence level (c) is 0.90, the sample mean (x̅) is 8.21, the population standard deviation (σ) is 0.62, and the sample size (n) is 8.
Step 2: Determine the appropriate distribution to use. Since the population standard deviation (σ) is known and the sample size (n) is small (n < 30), use the standard normal (Z) distribution to construct the confidence interval.
Step 3: Find the critical value (Z*) corresponding to the confidence level of 0.90. To do this, calculate the area in each tail as (1 - c) / 2 = (1 - 0.90) / 2 = 0.05. Use a Z-table or statistical software to find the Z* value that leaves 0.05 in each tail.
Step 4: Calculate the margin of error (E) using the formula: E = Z^* \(\cdot\) \(\frac{\sigma}{\sqrt{n}\)}. Substitute the values for Z*, σ, and n into the formula.
Step 5: Construct the confidence interval for the population mean using the formula: (x̅ - E, x̅ + E). Substitute the values for x̅ and E to find the lower and upper bounds of the confidence interval.

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Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the population parameter with a specified level of confidence. For example, a 90% confidence interval suggests that if we were to take many samples and construct intervals in the same way, approximately 90% of those intervals would contain the true population mean.
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Introduction to Confidence Intervals

Normal Distribution

The normal distribution is a continuous probability distribution characterized by its bell-shaped curve, defined by its mean and standard deviation. When constructing confidence intervals, if the sample size is small (typically n < 30) and the population standard deviation is known, the t-distribution is often used instead of the normal distribution, especially when the underlying population is normally distributed.
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Using the Normal Distribution to Approximate Binomial Probabilities

Standard Error

The standard error (SE) measures the variability of the sample mean estimate of a population mean. It is calculated as the population standard deviation divided by the square root of the sample size (σ/√n). A smaller standard error indicates that the sample mean is a more accurate estimate of the population mean, which is crucial for constructing a reliable confidence interval.
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Calculating Standard Deviation
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Domanda del libro di testo

[APPLET] The annual earnings (in dollars) for 30 randomly selected locksmiths are shown below. Assume the population is normally distributed. (Adapted from Salary.com)


48,69446,85642,91261,67271,11254,861


69,45471,84159,75169,61254,28452,166


66,36048,16465,27235,25061,12765,397


58,92558,91659,01753,07045,19969,941


69,49257,08553,82952,69268,29853,792



Construct a 95% confidence interval for the population mean annual earnings for locksmiths.

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Domanda del libro di testo

In Exercises 7–10, the statement represents a claim. Write its complement and state which is Ho and which is Ha.


σ=0.63

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Domanda del libro di testo

In Exercises 3–6, construct the indicated confidence interval for the population mean . Which distribution did you use to create the confidence interval?


c=0.95, x̅=3.46, s=1.63, n=16

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Domanda del libro di testo

The mean room rate for two adults for a random sample of 26 three-star hotels in Cincinnati has a sample standard deviation of \$31. Assume the population is normally distributed. (Adapted from Expedia)


Construct a 99% confidence interval for the population variance.

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An education organization claims that the mean SAT scores for male athletes and male non-athletes at a college are different. A random sample of 26 male athletes at the college has a mean SAT score of 1189 and a standard deviation of 218. A random sample of 18 male non-athletes at the college has a mean SAT score of 1376 and a standard deviation of 186. At α=0.05, can you support the organization’s claim? Interpret the decision in the context of the original claim. Assume the populations are normally distributed and the population variances are equal.

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Domanda del libro di testo

[APPLET] The annual earnings (in dollars) for 30 randomly selected locksmiths are shown below. Assume the population is normally distributed. (Adapted from Salary.com)


48,69446,85642,91261,67271,11254,861


69,45471,84159,75169,61254,28452,166


66,36048,16465,27235,25061,12765,397


58,92558,91659,01753,07045,19969,941


69,49257,08553,82952,69268,29853,792



A researcher claims that the mean annual earnings for locksmiths is \$55,000. At α=0.05, can you reject the researcher’s claim? Interpret the decision in the context of the original claim.

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