Chapter 11, Problem 1CRE

(a)

To determine

To construct: Graph for the distribution of grade levels for the sample data.

Explanation of Solution

Given:

Percentage of freshman $=29\%$

Percentage of sophomores $=27\%$

Percentage of juniors $=25\%$

Percentage of seniors $=19\%$

Number of freshman in a sample $=54$

Number of sophomores in a sample $=66$

Number of juniors in a sample $=56$

Number of seniors in a sample $=30$

Calculation:

Total number of students in a sample $=54+66+56+30=206$

Finding percentages:

Freshman: $\frac{54}{206}\times 100=26.21\%$

Sophomores: $\frac{66}{206}\times 100=32.05\%$

Juniors: $\frac{56}{206}\times 100=27.18\%$

Seniors: $\frac{30}{206}\times 100=14.56\%$

Graph:

For sample data

  

Interpretation:

From the above graph, it can be seen that most of the students are in sophomores group and then in juniors group, whereas, the claimed percentage says that most of the students lie in freshman group.

Hence, the statistics sample seems unusual.

(b)

To determine

To check: Whether the sample distribution differs from actual distribution or not.

Answer to Problem 1CRE

The sample distribution do not differ from actual distribution

Explanation of Solution

Formula used:

  ${\displaystyle \sum \frac{{\left(O-E\right)}^2}{E}~{\chi }_{n-1}}$

Calculation:

The expected count will be

	E
Freshman	$206\times 0.29=59.74$
Sophomores	$206\times 0.27=55.62$
Juniors	$206\times 0.25=51.5$
Seniors	$206\times 0.19=39.14$

The observed (O) and expected (E) counts are

	Expected (E)	Observed (O)
Freshman	$59.74$	$54$
Sophomores	$55.62$	$66$
Juniors	$51.5$	$56$
Seniors	$39.14$	$30$

The null and alternate hypotheses are

  $H_0:p_1=0.29,p_2=0.27,p_3=0.25,p_4=0.19$

  $H_1:$ At least one of the proportions is not equal.

Test statistic

  $\frac{{\left(54-59.74\right)}^2}{59.74}+\frac{{\left(66-55.62\right)}^2}{55.62}+\frac{{\left(56-51.5\right)}^2}{51.5}+\frac{{\left(30-39.14\right)}^2}{39.14}~{\chi }_3$

The test value is $5.016$

P value

  $P\left({\chi }_3>5.016\right)=0.170629$

Conclusion:

As the p value is greater than significance level, so there is insufficient evidence to reject null hypothesis at $5\%$ level of significance and thus conclude that there is no difference in actual and sample distribution and the sample data represents the actual data.