Data were collected on annual personal time (in hours) taken by a random sample of 16 women (group 1) and 7 men (group 2) employed by a medium-sized company. the women took an average of 24.75 hours of personal time per year with a standard deviation of 2.84 hours. the men took an average of 21.89 hours of personal time per year with a standard deviation of 3.29 hours. the human resources department believes that women tend to take more personal time than men because they tend to be the primary child caregivers in the family. the results of the test are t=2.02 with an associated p-value of 0.0352. the correct conclusion at α = 0.05 is to:

a. reject the null hypothesis.
b. fail to reject the null hypothesis.
c. conclude that women take a higher average number of hours of personal time per year compared to men.

After a dreary day of rain, the sun peeks through the clouds and a rainbow forms. you notice the rainbow is the shape of a parabola. the equation for this parabola is y = -x2 + 36. graph of a parabola opening down at the vertex 0 comma 36 crossing the x–axis at negative 6 comma 0 and 6 comma 0. in the distance, an airplane is taking off. as it ascends during take-off, it makes a slanted line that cuts through the rainbow at two points. create a table of at least four values for the function that includes two points of intersection between the airplane and the rainbow. analyze the two functions. answer the following reflection questions in complete sentences. what is the domain and range of the rainbow? explain what the domain and range represent. do all of the values make sense in this situation? why or why not? what are the x- and y-intercepts of the rainbow? explain what each intercept represents. is the linear function you created with your table positive or negative? explain. what are the solutions or solution to the system of equations created? explain what it or they represent. create your own piecewise function with at least two functions. explain, using complete sentences, the steps for graphing the function. graph the function by hand or using a graphing software of your choice (remember to submit the graph).