Chapter 7.1, Problem 13E

(a)

To determine

To find: the evidence that less than 60% of all students completed their assigned homework last week.

Explanation of Solution

Given:

  $\begin{array}{l}n=\text{100}\\ x=\text{45}\end{array}$

Formula used:

  $\widehat{p}=\frac{x}{n}$

Calculation:

Sample proportion is

  $\widehat{p}=\frac{x}{n}=\frac{45}{100}=0.45=45\%$

It is observed that the sample proportion 45% is less than the claimed 60% and therefore the sample proportion is evidence that the less than 60% of all students at the school completed their assigned homework last week.

(b)

To determine

To find: the two explanations for the evidence described in part (a).

Answer to Problem 13E

Sample proportion is 45% which is less than 60%

Explanation of Solution

Given:

  $\begin{array}{l}n=\text{100}\\ x=\text{45}\end{array}$

Formula used:

  $\widehat{p}=\frac{x}{n}$

Calculation:

Sample proportion is

  $\widehat{p}=\frac{x}{n}=\frac{45}{100}=0.45=45\%$

It is possible that the sample proportion is less than 60% the reason behind is that the population proportion is actually 60% and it is obtained a sample with a sample proportion of only 45% by chance.

Although, it is also possible that the sample proportion is less than 60% the reason behind is that the population proportion is actually less than 60%.

(c)

To determine

To Explain: the value represents.

Explanation of Solution

Given:

  

By seeing the dot plot it represents a sample proportion $\widehat{p}$ for a simple random sample with 100 students, where the sample proportion represents the proportion who completed their assigned homework last week in the sample.

The dot at 0.73 represents one simple random sample of students which 73 of the 100 students completed their assigned homework last week.

(d)

To determine

To find: that it is surprising to get a sample proportion of 0.45 or smaller in an SRS of size 100 when p = 0.60.

Answer to Problem 13E

Yes

Explanation of Solution

Given:

  

Each dot in the given dot plot represents a sample proportion $\widehat{p}$ for a simple random sample with 100 students, where the sample proportion represents the proportion who completed their assigned homework last week in the sample.

It is observed that there are no dots above 0.45 in the dot plot nor are there any dots to the left of 0.45.This then indicates that it is very unlikely that the sample proportion is $\widehat{p}=0.45$ when the population proportion is actual $p=0.60$ and therefore we would be surprised if obtained a sample with sample proportion $\widehat{p}=0.45$ .

(e)

To determine

To find: that is there convincing evidence that less than 60% of all students at the school completed their assigned homework last week.

Explanation of Solution

Given:

  

Each dot in the given dot plot represents a sample proportion $\widehat{p}$ for a simple random sample with 100 students, where the sample proportion represents the proportion who completed their assigned homework last week in the sample.

It is observed that there are no dots above 0.45 in the dot plot nor are there any dots to the left of 0.45.This then indicates that it is very unlikely that the sample proportion is $\widehat{p}=0.45$ when the population proportion is actual $p=0.60$ and therefore we would be surprised if obtained a sample with sample proportion $\widehat{p}=0.45$ .