Consider the following hypothesis test. H?: ? ? 0.55 H?: ? < 0.55 Compute the te

Question

Consider the following hypothesis test. H?: ? ? 0.55 H?: ? < 0.55 Compute the test statistic and the p-value for the following three cases. 14 n = 300 p? = 0.51 ? = 0.05 a p-value = 0.0823 Conclude that the population proportion is less than 0.55. b p-value = 0.0823 Conclude that the population proportion is not less than 0.55. c p-value = 0.0411 Conclude that the population proportion is not less than 0.55. d p-value = 0.0411 Conclude that the population proportion is less than 0.55. 15 n = 300 p? = 0.51 ? = 0.10 a p-value = 0.0411 Conclude that the population proportion is not less than 0.55. b p-value = 0.0411 Conclude that the population proportion is less than 0.55. c p-value = 0.0823 Conclude that the population proportion is not less than 0.55. d p-value = 0.0823 Conclude that the population proportion is less than 0.55. 16 n = 900 ?x = 468 a p-value = 0.0703 Reject H? at ? = 0.10, but do not reject at ? = 0.05. b p-value = 0.0703 Reject H? at ? = 0.05, but do not reject at ? = 0.10. c p-value = 0.0351 Reject H? at ? = 0.05, but do not reject at ? = 0.01. d p-value = 0.0351 Reject H? at ? = 0.10, but do not reject at ? = 0.05. Consider the following hypothesis test. H?: ? ? 0.55 H?: ? < 0.55 Compute the test statistic and the p-value for the following three cases. 14 n = 300 p? = 0.51 ? = 0.05 a p-value = 0.0823 Conclude that the population proportion is less than 0.55. b p-value = 0.0823 Conclude that the population proportion is not less than 0.55. c p-value = 0.0411 Conclude that the population proportion is not less than 0.55. d p-value = 0.0411 Conclude that the population proportion is less than 0.55. 15 n = 300 p? = 0.51 ? = 0.10 a p-value = 0.0411 Conclude that the population proportion is not less than 0.55. b p-value = 0.0411 Conclude that the population proportion is less than 0.55. c p-value = 0.0823 Conclude that the population proportion is not less than 0.55. d p-value = 0.0823 Conclude that the population proportion is less than 0.55. 16 n = 900 ?x = 468 a p-value = 0.0703 Reject H? at ? = 0.10, but do not reject at ? = 0.05. b p-value = 0.0703 Reject H? at ? = 0.05, but do not reject at ? = 0.10. c p-value = 0.0351 Reject H? at ? = 0.05, but do not reject at ? = 0.01. d p-value = 0.0351 Reject H? at ? = 0.10, but do not reject at ? = 0.05.

Explanation / Answer

The test hypothesis is

H?: ? ? 0.55 vs H?: ? < 0.55

The test statistic is

Z = (p? - ?)/sqrt(p?(1-p?)/n) ~ N(0,1) from Central limit theorem

In this case

Z = (0.51 - 0.55)/sqrt(0.51*0.49/300)

= -1.385918

P( Z < -1.385918) = 0.08288597 ~ 0.0823

Under level ? = 0.05, the hypothesis is not rejected

Answer:

The test statistic is

Z = (p? - ?)/sqrt(p?(1-p?)/n) ~ N(0,1) from Central limit theorem

In this case

Z = (0.51 - 0.55)/sqrt(0.51*0.49/300)

= -1.385918

P( Z < -1.385918) = 0.08288597 ~ 0.0823

Under level ? = 0.1, the hypothesis is rejected

Answer:

p? = ?x/n = 468/900

= 0.52

The test statistic is

Z = (p? - ?)/sqrt(p?(1-p?)/n) ~ N(0,1) from Central limit theorem

In this case

Z = (0.52 - 0.55)/sqrt(0.52*0.48/900)

= -1.801442

P(Z <  -1.040063) = 0.03581662

p-value = 0.03581662 ~ 0.0351

14 n = 300 p? = 0.51 ? = 0.05

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