Symposium is part of a larger work referred to as Plato's Dialogues. Wishart and Leach† found that about 21.4% of five-syllable sequences in Symposium are of the type in which four are short and one is long. Suppose an antiquities store in Athens has a very old manuscript that the owner claims is part of Plato's Dialogues. A random sample of 490 five-syllable sequences from this manuscript showed that 128 were of the type four short and one long. Do the data indicate that the population proportion of this type of five-syllable sequence is higher than that found in Plato's Symposium? Use ???? = 0.01.
(a)
What is the level of significance?
State the null and alternate hypotheses. (Enter != for ≠ as needed.)
H0:
p=0.214
H1:
p>0.214
(b)
What sampling distribution will you use? Do you think the sample size is sufficiently large? Explain.
We'll use the standard normal, because the sample size is not sufficiently large since np < 5 and nq < 5.We'll use the Student's t, because the sample size is not sufficiently large since np < 5 and nq < 5. We'll use the Student's t, because the sample size is sufficiently large since np > 5 and nq > 5.We'll use the standard normal, because the sample size is sufficiently large since np > 5 and nq > 5.
Use the value of the sample test statistic to find the corresponding z value. (Round your answer to two decimal places.)
(c)
Find the P-value of the test statistic. (Round your answer to four decimal places.)
P-value =
Sketch the sampling distribution and show the area corresponding to the P-value.
A graph has a horizontal axis with values from −3.5 to 3.5. The curve enters the window from the left, just above the horizontal axis, goes up and to the right, changes direction over approximately 0 on the horizontal axis, and then goes down and to the right before exiting the window just above the horizontal axis. The area under the curve to the right of 2.6 is shaded.
A graph has a horizontal axis with values from −3.5 to 3.5. The curve enters the window from the left, just above the horizontal axis, goes up and to the right, changes direction over approximately 0 on the horizontal axis, and then goes down and to the right before exiting the window just above the horizontal axis. The area under the curve to the left of 1.6 is shaded.
A graph has a horizontal axis with values from −3.5 to 3.5. The curve enters the window from the left, just above the horizontal axis, goes up and to the right, changes direction over approximately 0 on the horizontal axis, and then goes down and to the right before exiting the window just above the horizontal axis. The area under the curve to the left of −1.6 as well as the area under the curve to the right of 1.6 are both shaded.
A graph has a horizontal axis with values from −3.5 to 3.5. The curve enters the window from the left, just above the horizontal axis, goes up and to the right, changes direction over approximately 0 on the horizontal axis, and then goes down and to the right before exiting the window just above the horizontal axis. The area under the curve to the left of −1.6 is shaded.
(d)
Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level ?????
At the ???? = 0.01 level, we reject the null hypothesis and conclude the data are statistically significant.At the ???? = 0.01 level, we reject the null hypothesis and conclude the data are not statistically significant. At the ???? = 0.01 level, we fail to reject the null hypothesis and conclude the data are statistically significant.At the ???? = 0.01 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.
(e)
Interpret your conclusion in the context of the application.
There is sufficient evidence at the 0.01 level to conclude that the true proportion of the five-syllable sequence is higher than that in Plato's Symposium.There is insufficient evidence at the 0.01 level to conclude that the true proportion of the five-syllable sequence is higher than that in Plato's Symposium.