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Ch. 10 - Chi-Square Tests and the F-Distribution
Larson - Elementary Statistics: Picturing the World 8th Edition
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Larson 8th EditionCh. 10 - Chi-Square Tests and the F-Distribution문제 10.Q.4e
10장, 문제 10.Q.4e

In each exercise,
e. interpret the decision in the context of the original claim.
[APPLET] In Exercises 3 and 4, use the data, which list the annual wages (in thousands of dollars) for randomly selected individuals from three metropolitan areas. Assume the wages are normally distributed and that the samples are independent. (Adapted from U.S. Bureau of Economic Analysis)
Ithaca, NY: 53.0, 60.3, 34.6, 37.1, 46.6, 46.8, 41.4, 50.6, 50.8, 49.4, 35.0, 36.7, 57.1
Little Rock, AR: 50.7, 43.7, 53.4, 40.0, 45.2, 52.7, 35.2, 60.4, 40.0, 45.9, 45.7, 47.3, 46.5, 44.5, 31.5
Madison, WI: 62.4, 53.9, 67.6, 52.9, 67.7, 50.7, 62.1, 58.9, 61.1, 65.0, 60.4, 59.6, 51.3, 44.8, 66.2
Are the mean annual wages the same for all three cities? Use α=0.10. Assume that the population variances are equal.

검증된 단계별 안내
1
Step 1: Define the hypotheses for the ANOVA test. The null hypothesis \(H_0\) is that the mean annual wages are the same for all three cities: \(\mu_{Ithaca} = \mu_{Little\ Rock} = \mu_{Madison}\). The alternative hypothesis \(H_a\) is that at least one city has a different mean wage.
Step 2: Calculate the sample means and sample variances for each city using the given wage data. This involves summing the wages for each city and dividing by the number of observations to get the means, and then computing the variance for each city.
Step 3: Compute the overall mean wage by combining all the data from the three cities. This is done by summing all wages from all cities and dividing by the total number of observations.
Step 4: Calculate the between-group variability (Sum of Squares Between, SSB) and the within-group variability (Sum of Squares Within, SSW). Use the formulas: \[SSB = \sum_{i=1}^k n_i (\bar{x}_i - \bar{x})^2\] \[SSW = \sum_{i=1}^k \sum_{j=1}^{n_i} (x_{ij} - \bar{x}_i)^2\] where \(k\) is the number of groups (3 cities), \(n_i\) is the sample size for group \(i\), \(\bar{x}_i\) is the sample mean for group \(i\), and \(\bar{x}\) is the overall mean.
Step 5: Calculate the Mean Squares Between (MSB) and Mean Squares Within (MSW) by dividing the sums of squares by their respective degrees of freedom: \[MSB = \frac{SSB}{k - 1}\] \[MSW = \frac{SSW}{N - k}\] where \(N\) is the total number of observations. Then compute the F-statistic: \[F = \frac{MSB}{MSW}\] Compare this F-statistic to the critical value from the F-distribution with \((k-1, N-k)\) degrees of freedom at the \(\alpha = 0.10\) significance level to decide whether to reject or fail to reject the null hypothesis.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Analysis of Variance (ANOVA)

ANOVA is a statistical method used to compare the means of three or more groups to determine if at least one group mean differs significantly. It tests the null hypothesis that all group means are equal by analyzing variance within and between groups. This method is appropriate here since we compare mean wages across three cities.
추천 영상:
가이드 코스
06:46
Introduction to ANOVA

Assumption of Equal Population Variances

ANOVA assumes that the populations from which samples are drawn have equal variances (homogeneity of variance). This assumption ensures the validity of the F-test used in ANOVA. If variances differ greatly, the test results may be unreliable, so verifying or assuming equal variances is crucial for this analysis.
추천 영상:
가이드 코스
04:48
Population Standard Deviation Known

Interpreting the Decision in Context

After conducting ANOVA, interpreting the decision involves relating the statistical conclusion back to the original claim about mean wages. If the null hypothesis is rejected, it means at least one city's mean wage differs significantly. If not rejected, it suggests no evidence of wage differences among the cities at the chosen significance level.
추천 영상:
가이드 코스
07:10
Empirical Rule of Standard Deviation and Range Rule of Thumb
관련 실천
교과서 질문

In each exercise,

c. find the test statistic,

[APPLET] In Exercises 3 and 4, use the data, which list the annual wages (in thousands of dollars) for randomly selected individuals from three metropolitan areas. Assume the wages are normally distributed and that the samples are independent. (Adapted from U.S. Bureau of Economic Analysis)

Ithaca, NY: 53.0, 60.3, 34.6, 37.1, 46.6, 46.8, 41.4, 50.6, 50.8, 49.4, 35.0, 36.7, 57.1

Little Rock, AR: 50.7, 43.7, 53.4, 40.0, 45.2, 52.7, 35.2, 60.4, 40.0, 45.9, 45.7, 47.3, 46.5, 44.5, 31.5

Madison, WI: 62.4, 53.9, 67.6, 52.9, 67.7, 50.7, 62.1, 58.9, 61.1, 65.0, 60.4, 59.6, 51.3, 44.8, 66.2

Are the mean annual wages the same for all three cities? Use α=0.10. Assume that the population variances are equal.

25
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교과서 질문

In each exercise,

c. find the test statistic,


In Exercises 1 and 2, use the table, which lists the distribution of educational achievement for people in the United States ages 25 and older. It also lists the results of a random survey for two additional age groups. (Adapted from U.S. Census Bureau)



Use the data for 30- to 34-year-olds and 65- to 69-year-olds to test whether age and educational attainment are related. Use α=0.01.

42
views
교과서 질문

In each exercise,

d. decide whether to reject or fail to reject the null hypothesis, and

[APPLET] In Exercises 3 and 4, use the data, which list the annual wages (in thousands of dollars) for randomly selected individuals from three metropolitan areas. Assume the wages are normally distributed and that the samples are independent. (Adapted from U.S. Bureau of Economic Analysis)

Ithaca, NY: 53.0, 60.3, 34.6, 37.1, 46.6, 46.8, 41.4, 50.6, 50.8, 49.4, 35.0, 36.7, 57.1

Little Rock, AR: 50.7, 43.7, 53.4, 40.0, 45.2, 52.7, 35.2, 60.4, 40.0, 45.9, 45.7, 47.3, 46.5, 44.5, 31.5

Madison, WI: 62.4, 53.9, 67.6, 52.9, 67.7, 50.7, 62.1, 58.9, 61.1, 65.0, 60.4, 59.6, 51.3, 44.8, 66.2

Are the mean annual wages the same for all three cities? Use α=0.10. Assume that the population variances are equal.

43
views
교과서 질문

In each exercise,

a. identify the claim and state H₀ and Hₐ,


In Exercises 1 and 2, use the table, which lists the distribution of educational achievement for people in the United States ages 25 and older. It also lists the results of a random survey for two additional age groups. (Adapted from U.S. Census Bureau)


Use the data for 30- to 34-year-olds and 65- to 69-year-olds to test whether age and educational attainment are related. Use α=0.01.

43
views
교과서 질문

In each exercise,

b. find the critical value and identify the rejection region,


In Exercises 1 and 2, use the table, which lists the distribution of educational achievement for people in the United States ages 25 and older. It also lists the results of a random survey for two additional age groups. (Adapted from U.S. Census Bureau)


Use the data for 30- to 34-year-olds and 65- to 69-year-olds to test whether age and educational attainment are related. Use α=0.01.

38
views
교과서 질문

In each exercise,

d. decide whether to reject or fail to reject the null hypothesis, and

e. interpret the decision in the context of the original claim.


In Exercises 1 and 2, use the table, which lists the distribution of educational achievement for people in the United States ages 25 and older. It also lists the results of a random survey for two additional age groups. (Adapted from U.S. Census Bureau)



Use the data for 30- to 34-year-olds and 65- to 69-year-olds to test whether age and educational attainment are related. Use α=0.01.

70
views