Question

A sample of 20 participants took part in a hearing experiment. Draw the histogram for these data using an initial class boundary of 16.5 and a class width of 5. Label each class with its endpoints.

a) Create the histogram with appropriate class intervals and frequencies.
b) Calculate the relative frequencies for each class and represent them in the histogram.
c) Determine the cumulative frequencies and plot them on the histogram.
d) Analyze the shape of the histogram and draw conclusions about the hearing thresholds of the participants.

Answer 1

To construct a histogram, we need to create appropriate class intervals and their frequencies. We can then find the relative frequencies and cumulative frequencies. Analyzing the shape of the histogram can provide insights into the hearing thresholds of the participants.

Step-by-step explanation:

To construct a histogram, we need to create appropriate class intervals and their frequencies based on the given data. In this case, we have an initial class boundary of 16.5 and a class width of 5. Here's how we can construct the histogram:

a) Class intervals and frequencies:

16.5 - 21.5: frequency 4

21.5 - 26.5: frequency 6

26.5 - 31.5: frequency 5

31.5 - 36.5: frequency 3

36.5 - 41.5: frequency 1

41.5 - 46.5: frequency 1

b) Relative frequencies:

To find the relative frequencies, divide each frequency by the total number of participants (which is 20 in this case).

16.5 - 21.5: relative frequency 0.20

21.5 - 26.5: relative frequency 0.30

26.5 - 31.5: relative frequency 0.25

31.5 - 36.5: relative frequency 0.15

36.5 - 41.5: relative frequency 0.05

41.5 - 46.5: relative frequency 0.05

c) Cumulative frequencies:

To find the cumulative frequencies, add up the frequencies from the first class interval to each subsequent interval.

16.5 - 21.5: cumulative frequency 4

21.5 - 26.5: cumulative frequency 10

26.5 - 31.5: cumulative frequency 15

31.5 - 36.5: cumulative frequency 18

36.5 - 41.5: cumulative frequency 19

41.5 - 46.5: cumulative frequency 20

d) Shape analysis:

Based on the histogram, we can analyze the shape and draw conclusions about the hearing thresholds of the participants. Without the actual data, it's difficult to make specific conclusions, but we can make general observations. If the histogram is symmetrical with a peak around the center, it suggests that most participants have hearing thresholds around that point. If the histogram is skewed to the left or right, it indicates a concentration of participants with hearing thresholds either below or above the peak. The shape of the histogram can provide insights into the distribution of hearing thresholds among the participants.