Question

A random sample of 40 binomial trials resulted in 16 successes. Test the claim that the population proportion of successes does not equal 0.50. Use a level of significance of 0.05.

(a) Can a normal distribution be used for the *p̂*
distribution? Explain.

No, *nq* is greater than 5, but *np* is less than
5.Yes, *np* and *nq* are both greater than
5. No, *np* is greater than 5, but
*nq* is less than 5.No, *np* and *nq* are both
less than 5.Yes, *np* and *nq* are both less than
5.

(b) State the hypotheses.

*H*_{0}: *p* = 0.5;
*H*_{1}: *p* > 0.5*H*_{0}:
*p* = 0.5; *H*_{1}: *p* ≠
0.5 *H*_{0}: *p* =
0.5; *H*_{1}: *p* <
0.5*H*_{0}: *p* < 0.5;
*H*_{1}: *p* = 0.5

(c) Compute *p̂*.

Compute the corresponding standardized sample test statistic.
(Round your answer to two decimal places.)

(d) Find the *P*-value of the test statistic. (Round your
answer to four decimal places.)

(e) Do you reject or fail to reject

*H*_{0}?

Explain.

At the *α* = 0.05 level, we reject the null hypothesis
and conclude the data are statistically significant.At the
*α* = 0.05 level, we reject the null hypothesis and conclude
the data are not statistically
significant. At the *α* = 0.05 level,
we fail to reject the null hypothesis and conclude the data are
statistically significant.At the *α* = 0.05 level, we fail
to reject the null hypothesis and conclude the data are not
statistically significant.

(f) What do the results tell you?

The sample *p̂* value based on 40 trials is sufficiently
different from 0.50 to justify rejecting *H*_{0} for
*α* = 0.05.The sample *p̂* value based on 40 trials
is not sufficiently different from 0.50 to not reject
*H*_{0} for *α* =
0.05. The sample *p̂* value based on
40 trials is sufficiently different from 0.50 to not reject
*H*_{0} for *α* = 0.05.The sample *p̂*
value based on 40 trials is not sufficiently different from 0.50 to
justify rejecting *H*_{0} for *α* = 0.05.

Answer #1

The statistical software output for this problem is:

Hence,

a) Yes, *np* and *nq* are both greater than 5.

b) *H*_{0}: *p* = 0.5;
*H*_{1}: *p* ≠ 0.5

c) p = 0.4

Test statistic = -1.26

d) P - value = 0.2059

e) **Option D** is correct.

f) **Option B** is correct.

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