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= 19inches. However, a survey reported that of a random sample of51fish caught, the mean length wasx=18.6inches, with estimated standard deviations=2.9inches. Do these data indicate that the average length of a trout caught in Pyramid Lake is less than= 19inches? Use= 0.05.

(a)What is the level of significance?

State the null and alternate hypotheses.

H₀:= 19 in;H₁:> 19 in

H₀:= 19 in;H₁:19 in

H₀:< 19 in;H₁:= 19 in

H₀:> 19 in;H₁:= 19 in

H₀:= 19 in;H₁:< 19 in

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

The standard normal, since the sample size is large andis known.

The Student'st, since the sample size is large andis known.

The standard normal, since the sample size is large andis unknown.

The Student'st, since the sample size is large andis unknown.

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

(c)Estimate theP-value.

P-value > 0.010

0.0010 <P-value < 0.010

0.250 <P-value < 0.0010

0.125 <P-value < 0.250

P-value < 0.125

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

A plot of the Student's t-probability curve has a horizontal axis with values from4 to 4. 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 between4 and0.99 is shaded.

A plot of the Student's t-probability curve has a horizontal axis with values from4 to 4. 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 between0.99 and 4 is shaded.

A plot of the Student's t-probability curve has a horizontal axis with values from4 to 4. 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 between4 and0.99 as well as the area under the curve between 0.99 and 4 are both shaded.

A plot of the Student's t-probability curve has a horizontal axis with values from4 to 4. 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 between 0.99 and 4 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= 0.05 level, we reject the null hypothesis and conclude the data are statistically significant.

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

At the= 0.05 level, we fail to reject the null hypothesis and conclude the data are statistically significant.

At the= 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.