In: Statistics and Probability
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 489 five-syllable sequences from this
manuscript showed that 129 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 α = 0.01.
(a) What is the level of significance?
State the null and alternate hypotheses.
H0: p > 0.214; H1: p = 0.214
H0: p = 0.214; H1: p > 0.214
H0: p = 0.214; H1: p < 0.214
H0: p < 0.214; H1: p = 0.214
H0: p = 0.214; H1: p ≠ 0.214
(b) What sampling distribution will you use?
The standard normal, since np > 5 and nq > 5.
The standard normal, since np < 5 and nq < 5.
The Student's t, since np > 5 and nq > 5.
The Student's t, since np < 5 and nq < 5.
What is the value of the sample test statistic? (Round your answer to two decimal places.)
(c) Find the P-value of the test statistic. (Round your answer to
four decimal places.)
Sketch the sampling distribution and show the area corresponding to
the P-value.
WebAssign Plot WebAssign Plot
WebAssign Plot WebAssign Plot
(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 α?
At the α = 0.01 level, we reject the null hypothesis and conclude
the data are statistically significant.
At the α = 0.01 level, we reject the null hypothesis and conclude
the data are not statistically significant.
At the α = 0.01 level, we fail to reject the null hypothesis and
conclude the data are statistically significant.
At the α = 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.
n= 489, x= 129, P = 21.4% = 0.214, = 0.01
a)
level of significance = 0.01
The null and alternative hypothesis is
Ho: P = 0.214
H1: P > 0.214
b)
The standard normal, since np
> 5 and nq > 5.
formula for test statistics is
test statistics: z =2.69
c)
calculate P-Value
P-Value = 1 - P(z < 2.69)
using z table we get
P(z < 2.69) = 0.9964
P-Value = 1 - 0.9964
P-Value = 0.0036
d)
decision rule is
Reject Ho if ( P-value ) ( )
here, ( P-value= 0.0036 ) < ( = 0.01)
Hence, we can say,
Null hypothesis is rejected.
At the α = 0.01 level, we reject the null hypothesis and conclude the data are statistically significant.
e)
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.