Question

Pyramid Lake is on the Paiute Indian Reservation in Nevada. The lake is famous for cutthroat trout. Suppose a friend tells you that the average length of trout caught in Pyramid Lake is μ = 19 inches. However, a survey reported that of a random sample of 51 fish caught, the mean length was x = 18.6 inches, with estimated standard deviation s = 3.1 inches. Do these data indicate that the average length of a trout caught in Pyramid Lake is less than μ = 19 inches? Use α = 0.05.

(a) What is the level of significance?

State the null and alternate hypotheses.

H₀: μ = 19 in; H₁: μ ≠ 19 inH₀: μ = 19 in; H₁: μ < 19 in     H₀: μ = 19 in; H₁: μ > 19 inH₀: μ > 19 in; H₁: μ = 19 inH₀: μ < 19 in; H₁: μ = 19 in

(b) What sampling distribution will you use? Explain the rationale for your choice of sampling distribution.

The Student's t, since the sample size is large and σ is unknown.The standard normal, since the sample size is large and σ is unknown.    The Student's t, since the sample size is large and σ is known.The standard normal, since the sample size is large and σ is known.

What is the value of the sample test statistic? (Round your answer to three decimal places.)

(c) Estimate the P-value.

P-value > 0.2500.100 < P-value < 0.250    0.050 < P-value < 0.1000.010 < P-value < 0.050P-value < 0.010

Sketch the sampling distribution and show the area corresponding to the P-value.

(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 = 0.05 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.

(e) Interpret your conclusion in the context of the application.

There is sufficient evidence at the 0.05 level to conclude that the average fish length is less than 19 inches.There is insufficient evidence at the 0.05 level to conclude that the average fish length is less than 19 inches.    

Answer 1

using excel>addin>phstat>one sample test

we have

t Test for Hypothesis of the Mean
Data
Null Hypothesis                m=	19
Level of Significance	0.05
Sample Size	51
Sample Mean	18.6
Sample Standard Deviation	3.1
Intermediate Calculations
Standard Error of the Mean	0.4341
Degrees of Freedom	50
t Test Statistic	-0.9215
Lower-Tail Test
Lower Critical Value	-1.6759
p-Value	0.1806
Do not reject the null hypothesis

(a) the level of significance is 0.05

State the null and alternate hypotheses.

H₀: μ = 19 in; H₁: μ < 19

(b) What sampling distribution will you use? Explain the rationale for your choice of sampling distribution.

The Student's t, since the sample size is large and σ is unknown.

What is the value of the sample test statistic? -0.922

(c) Estimate the P-value.

0.100 < P-value < 0.250  

(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 = 0.05 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 insufficient evidence at the 0.05 level to conclude that the average fish length is less than 19 inches.