Question

part 1: practice with the big ideas
as a college student, you are working hard and trying to decide what type of career path will be the best fit for you. will you find a path that you will stay on, or will you decide at some point to take a different path? a claim is made that 50% of all adults decide at some point to change careers. a researcher believes this value is not high enough. upon surveying a random sample of 380 adults, the researcher finds that 209 of these adults have changed careers. the researcher would like to use the sample data to conduct a hypothesis test at a significance level of 0.05.
use the above information to answer questions 1 through 8.

1. what is ( p ) in this problem? tell us in words what this is and give us the numerical value of ( p ).
2. of the 380 adults who were surveyed, 209 have changed careers. use this information to compute a sample proportion in the space below.
3. what are the null and alternative hypotheses in this example? please write or type them below, using appropriate symbols, signs, and numbers.

( h_{0} ):
( h_{a} ):

Explanation:

Step1: Define \( p \)

\( p \) is the population proportion. In this context, it represents the proportion of all adults who decide to change careers. The claim is that \( p = 0.50 \).

Step2: Calculate sample proportion \( \hat{p} \)

The formula for sample proportion is \( \hat{p}=\frac{x}{n} \), where \( x = 209 \) (number of adults who changed careers) and \( n=380 \) (sample size).

$$ \hat{p}=\frac{209}{380}=0.55 $$

Step3: Formulate hypotheses

The null hypothesis \( H_{0} \) is the statement of no - change or the status - quo. The alternative hypothesis \( H_{a} \) is the researcher's claim.
Since the researcher believes the proportion is higher than the claimed \( 50\% \),
\( H_{0}:p = 0.50 \)
\( H_{a}:p>0.50 \)

Answer:

1. \( p \) is the proportion of all adults who decide to change careers. Numerically, \( p = 0.50 \).
2. The sample proportion \( \hat{p}=\frac{209}{380}=0.55 \).
3. \( H_{0}:p = 0.50 \); \( H_{a}:p>0.50 \)