Question

5. A certain size bag is designed to hold 25 pounds of potatoes. A farmer fills such bags in the field. Assume that the weight Y of potatoes in a bag follows N(u,9). We would like to test Ho 25 vs. H <25. Let Yi, Y2, Y, Y, be a random sample of size 4 from this distribution, and let the rejection region for this test is Y 22.5 (a) What is the significance level of this test? (b) What is the power of this test when =23? (e) Suppose we observe y associated with this sample? 22, 24.5, -23, 26.5, What is the P-value

Answer 1

Mean weight of potatoes in a bag: Sample standard deviation of the fills with the data normally distributed N(u,9) Y_:=22.5 a= V9=3 Null hypothesis: Но и-25 Alternative hypothesis: Н, и<25 a Sample size: n:=4 Degrees of freedom: d:=n-1 3 a. Level of significance of test: a:=0.05

y__H-1.667 Z test statistic for the mean less than 22.5 Z stat Vn For 23, Zstat -0.333 -0.333 2 (1}. {z-p 1 dz 0.3696 power= b. Power of the test: V2.T.o -6 z score for lower tail critical region of a standard normal distribution for a 95% confidence level, is 0.025, 0.025 from reference table. 2 Z =-1.6449 Lower tail critical value: 2 P- value: p 0.0478

is less than the level of Because Zgtat=-1.667 < Z ,and the p value p 0.048 significance test 0.05, so you reject HO. You have insufficient evidence to conclude that the mean weight of the bags filled with 25 pounds of potatoes, and it is not equal to 25 pounds but it is less than 25 pounds. y 3 Συ, 22 i-0 Y_= 24 24.5 23 26.5 Y_-P__0.667 Zstat c. Z test statistic for the mean less than 24 σ

-0.667 _( 1 }, { z-p\ 1 p value: dz 0.2524 pvalue .e V2.7T.T -6

A C E F Z Test of Hypothesis for the Mean 1 2 B Data 3 Z Test of Hypothesis for the Mean 1 4 Null Hypothesis 5 Level of Significance 6 Population Standard Deviation Sample Size Sample Mean 25 0.05 2 Data Null Hypothesis Level of Significance Population Standard Deviation Sample Size Sample Mean 25 4 3 o.05 4 22.5 7 6 4 7 22.5 8 9 Intermediate Calculations Intermediate Calculations 10 10 Standard Error of the Mean Z Test Statistic 1.5 11 Standard Error of the Mean -B6 /SQRT (B7) |-(B8-B4)/B11 11 1.666666667 12 12 Z Test Statistic 13 Lower-Tail Test 13 14 Lower Critical Value 15 1.644853627 Lower-Tail Test 14 0.047790352 16 P-Value =NORMSINV(B5) =NORMSDIST(B12) 15 Lower Critical Value Reject the null hypothesis 17 16 p-Value 17 IF(B16<SB$5,"Reject the null hypothesis", 18 18 19 LONCO On