In: Statistics and Probability
Question 1-
A graphing calculator is recommended.
Are the mean number of times a month a person eats out the same for
whites, blacks, Hispanics and Asians? Suppose that the table below
shows the results of a study.
White | Black | Hispanic | Asian |
---|---|---|---|
5 | 3 | 8 | 7 |
8 | 1 | 3 | 3 |
2 | 5 | 5 | 5 |
4 | 2 | 4 | 1 |
6 | 6 | 7 |
Assume that all distributions are normal, the four population standard deviations are approximately the same, and the data were collected independently and randomly. Use a level of significance of 0.05. (Let 1 = White, 2 = Black, 3 = Hispanic, and 4 = Asian.)
1) Enter an exact number as an integer, fraction, or decimal.
df(num) =
2)Enter an exact number as an integer, fraction, or decimal.
df(denom) =
3)What is the test statistic? (Round your answer to two decimal places.)
4) What is the p-value? (Round your answer to four decimal places.)
5) Alpha (Enter an exact number as an integer, fraction, or decimal.)
α =
Question 2-College students may be interested in whether or not their majors have any effect on starting salaries after graduation. Suppose that 298 recent graduates were surveyed as to their majors in college and their starting salaries after graduation. Below are the data. Conduct a test of independence. (Use a significance level of 0.05.)
Major | < $50,000 | $50,000 - $68,999 | $69,000 + |
---|---|---|---|
English | 4 | 20 | 4 |
Engineering | 10 | 30 | 61 |
Nursing | 10 | 16 | 14 |
Business | 9 | 20 | 31 |
Psychology | 19 | 29 | 21 |
1) What is the test statistic? (Round your answer to two decimal places.)
2) Alpha (Enter an exact number as an integer, fraction, or
decimal.)
α =
A | B | C | D | |||
count, ni = | 5 | 4 | 5 | 5 | ||
mean , x̅ i = | 5.000 | 2.75 | 5.20 | 4.60 | ||
std. dev., si = | 2.236 | 1.708 | 1.924 | 2.61 | ||
sample variances, si^2 = | 5.000 | 2.917 | 3.700 | 6.800 | ||
total sum | 25 | 11 | 26 | 23 | 85 | (grand sum) |
grand mean , x̅̅ = | Σni*x̅i/Σni = | 4.47 | ||||
square of deviation of sample mean from grand mean,( x̅ - x̅̅)² | 0.277 | 2.971 | 0.528 | 0.016 | ||
TOTAL | ||||||
SS(between)= SSB = Σn( x̅ - x̅̅)² = | 1.385 | 11.884 | 2.638 | 0.080 | 15.98684 | |
SS(within ) = SSW = Σ(n-1)s² = | 20.000 | 8.750 | 14.800 | 27.200 | 70.7500 |
no. of treatment , k = 4
df between = k-1 = 3
N = Σn = 19
df within = N-k = 15
mean square between groups , MSB = SSB/k-1 =
5.3289
mean square within groups , MSW = SSW/N-k =
4.7167
F-stat = MSB/MSW = 1.13
anova table | ||||||
SS | df | MS | F | p-value | F-critical | |
Between: | 15.99 | 3 | 5.33 | 1.13 | 0.3686 | 3.29 |
Within: | 70.75 | 15 | 4.72 | |||
Total: | 86.74 | 18 | ||||
α = | 0.05 | |||||
conclusion : | p-value>α , do not reject null hypothesis |
P VALUE = 0.3686
ALPHA = 0.05
...................
2)
Observed Frequencies | |||||||
0 | |||||||
0 | <50000 | 50000-68999 | 69000+ | Total | |||
ENG | 4 | 20 | 4 | 28 | |||
ENGGRNG | 10 | 30 | 61 | 101 | |||
NURSING | 10 | 16 | 14 | 40 | |||
BUSINESS | 9 | 20 | 31 | 60 | |||
PSYCHOLOGY | 19 | 29 | 21 | 69 | |||
Total | 52 | 115 | 131 | 298 | |||
Expected frequency of a cell = sum of row*sum of column / total sum | |||||||
Expected Frequencies | |||||||
<50000 | 50000-68999 | 69000+ | Total | ||||
ENG | 52*28/298=4.886 | 115*28/298=10.805 | 131*28/298=12.309 | 28 | |||
ENGGRNG | 52*101/298=17.624 | 115*101/298=38.977 | 131*101/298=44.399 | 101 | |||
NURSING | 52*40/298=6.98 | 115*40/298=15.436 | 131*40/298=17.584 | 40 | |||
BUSINESS | 52*60/298=10.47 | 115*60/298=23.154 | 131*60/298=26.376 | 60 | |||
PSYCHOLOGY | 52*69/298=12.04 | 115*69/298=26.628 | 131*69/298=30.332 | 69 | |||
Total | 52 | 115 | 131 | 298 | |||
(fo-fe)^2/fe | |||||||
ENG | 0.161 | 7.824 | 5.609 | ||||
ENGGRNG | 3.298 | 2.067 | 6.207 | ||||
NURSING | 1.3068 | 0.0206 | 0.7305 | ||||
BUSINESS | 0.2063 | 0.4297 | 0.8107 | ||||
PSYCHOLOGY | 4.0230 | 0.2114 | 2.8712 |
Chi-Square Test Statistic,χ² = Σ(fo-fe)^2/fe
= 35.78
Level of Significance =
0.05
Number of Rows = 5
Number of Columns = 3
Degrees of Freedom=(#row - 1)(#column -1) = (5- 1 ) * ( 3- 1 )
= 8
p-Value = 0.0000 [Excel function:
=CHISQ.DIST.RT(χ²,df) ]
Decision: p-value < α , Reject Ho
..................
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