Ch. 6 - Confidence Intervals

Larson - Elementary Statistics: Picturing the World 8th Edition

Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook

Larson 8th Edition

Ch. 6 - Confidence Intervals

Problem 6.2.23

Chapter 6, Problem 6.2.23

You research prices of cell phones and find that the population mean is \$431.61. In Exercise 19, does the t-value fall between -t0.95 and t0.95?

Verified step by step guidance

Step 1: Understand the problem. The question is asking whether the calculated t-value falls within the range defined by the critical t-values (-t₀.₉₅ and t₀.₉₅) for a given confidence level (95%). This involves hypothesis testing and comparing the t-value to the critical values.

Step 2: Identify the formula for the t-value. The t-value is calculated using the formula:

t = \frac{x - μ}{\frac{s}{\sqrt{n}}}

, where

x

is the sample mean,

μ

is the population mean,

s

is the sample standard deviation, and

n

is the sample size.

Step 3: Determine the critical t-values (-t₀.₉₅ and t₀.₉₅). These values depend on the degrees of freedom (df), which is calculated as

df = n - 1

. Use a t-distribution table or statistical software to find the critical values for a 95% confidence level.

Step 4: Calculate the t-value using the formula from Step 2. Plug in the sample mean, population mean, sample standard deviation, and sample size into the formula to compute the t-value.

Step 5: Compare the calculated t-value to the critical values (-t₀.₉₅ and t₀.₉₅). If the t-value falls within this range, it is within the 95% confidence interval. Otherwise, it is outside the interval.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Population Mean

The population mean is the average value of a set of data points in a complete population. It is calculated by summing all the values and dividing by the number of values. In this context, the population mean of $431.61 represents the average price of cell phones across the entire population being studied.

t-Distribution

The t-distribution is a type of probability distribution that is symmetric and bell-shaped, similar to the normal distribution but with heavier tails. It is used in statistics when the sample size is small or the population standard deviation is unknown. The t-values, such as -t0.95 and t0.95, represent critical values that define the boundaries for hypothesis testing.

Hypothesis Testing

Hypothesis testing is a statistical method used to make inferences about population parameters based on sample data. It involves formulating a null hypothesis and an alternative hypothesis, then using test statistics (like t-values) to determine whether to reject the null hypothesis. The comparison of the calculated t-value to critical values helps assess the significance of the results.

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06:21

Step 1: Write Hypotheses

Related Practice

Textbook Question

In Exercises 5–8, find the critical value zc necessary to construct a confidence interval at the level of confidence c.

c = 0.80

Textbook Question

Finding Critical Values for χ2 In Exercises 3–8, find the critical values χR2 and χL2 for the level of confidence c and sample size n.

c = 0.95, n = 20

Textbook Question

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Textbook Question

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Textbook Question

In Exercises 35–40, use the standard normal distribution or the t-distribution to construct a 95% confidence interval for the population mean. Justify your decision. If neither distribution can be used, explain why. Interpret the results.

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Textbook Question

In Exercises 25–28, use the confidence interval to find the margin of error and the sample mean.

(3.144, 3.176)