Question

A		C	D
11		211	211
12		125	121
7		179	185
12		225	222
11		161	157
15		170	174
6		191	184
16		195	194
12		135	133
13		162	165
9
14
11
10
8
15
14
13
9
6
8
12
14
16
11

1. The mean of the distribution of potato sack weights is 80 lbs with a standard deviation of 4 lbs. Assume that the distribution approximates a Bell curve.

1. Use the z-table calculator and find the weight which is the 70th percentile. Use value from an are (Show a screen shot for your answer.)
2. What is the z-score for a weight of 75 lbs? z-score = (x-µ)/σ
3. Suppose you pick a single sack at random. What is the probability that the weight will be between 85 and 100 lbs? Use area from a value. (Show a screen shot for your answer.)

Use the central limit theorem to the following questions.

4. Suppose you pick a group of 9 sacks instea What would be the standard deviation of the sample’s (group’s) average using the central limit theorem? (i.e. σ_xbar = σ/)
5. What is the probability that a group of 9 sacks will have an average weight between 90 and 100 lbs? Use the z-table calculator with “area from a value”. (Show a screen shot for your answer.)

2. Use Column A.

We want to test to see whether the data taken from 25 test experiments is consistent with the mean equal to 10.3 (µ = 10.3), or is more consistent with the mean greater than 10.3 (µ > 10.3).

Use Summary 5b, Table 2, Column 1.

1. What is the null hypothesis H_o for our test?
2. What is the alternative hypothesis H_a?
3. What type of tail test will we use? (left tail, right tail, or two tails)?
4. What is the mean of the sample xbar?
5. What is the standard deviation of the sample s?
6. What is the size of the sample n?
7. We going to use a t-statistic. Explain why we are not going to use a z-statistic.
8. Calculate the t-statistic using xbar, µ, n, and s.
9. How many degrees of freedom does this data set have?
10. Use the t-distribution calculator to compute a p-value. Include a screen shot of your answer.
11. We want a 99% confidence level. Based on your value of p, should we accept or reject the null hypothesis?
12. What if we want a 95% confidence level? Based on your value of p, should we accept or reject the null hypothesis?

3. Use Columns C and D for this question.

You are measuring weight loss using the same set of people at different times C and D. You want to know whether there is any difference in the weight between the start of the diet and the end of the diet. Column C gives the weight at the beginning of diet time. Column D gives the weight for the SAME person at the end of the diet time. Since there is data for the same person at different times, we will test whether µ(C-D) <= 0 or µ(C-D) > 0 (meaning the diet did cause weight loss) since we have correlated data (matched pairs).

Use Summary 5b, Table 2, Column 1

1. Make a new series of data samples by letting E = C – D. List your new series of 10 numbers.
2. What is the null hypothesis H₀ ?
3. What is the alternative hypothesis H_a ?
4. What type of tail test are we going to use? (left tail, right tail, two tail)
5. What is the mean xbar of this new sample?
6. What is the standard deviation of the sample s?
7. What is the size of the sample n?
8. How many degrees of freedom does this data set have?
9. What is the t-statistic for this sample?
10. Use the t-distribution calculator to compute a p-value. Show a screen shot of your answer.
11. Based on this value of p and using a 90% confidence level, is there a systematic difference in the weight changes between the start and stop time of the diet? Should we accept or reject the null hypothesis?

Answer 1

1. The mean of the distribution of potato sack weights is 80 lbs with a standard deviation of 4 lbs. Assume that the distribution approximates a Bell curve

a)

µ=   80                  
σ =    4                  
proportion=   0.7                  
                      
Z value at    0.7   =   0.52   (excel formula =NORMSINV(   0.7   ) )
z=(x-µ)/σ                      
so, X=zσ+µ=   0.52   *   4   +   80  
X   =   82.10      

b)

Z=(X-µ)/σ=   (75-80)/4)=      -1.25

c)

µ =    80                              
σ =    4                              
we need to calculate probability for ,                                  
P (   85   < X <   100   )                  
=P( (85-80)/4 < (X-µ)/σ < (100-80)/4 )                                  
                                  
P (    1.250   < Z <    5.000   )                   
= P ( Z <    5.000   ) - P ( Z <   1.250   ) =    1.0000   -    0.8944   =    0.1056

Use the central limit theorem to the following questions.

a)

S.D. = σ/ sqrt(n)  

= 4/ sqrt(9)  

= 4/3 = 1.33

b)

µ =    80                                  
σ =    4                                  
n=   9                                  
we need to calculate probability for ,                                      
90   ≤ X ≤    100                              
X1 =    90   ,    X2 =   100                      
                                      
Z1 =   (X1 - µ )/(σ/√n) = (   90   -   80   ) / (   4   / √   9   ) =   7.50
Z2 =   (X2 - µ )/(σ/√n) = (   100   -   80   ) / (   4   / √   9   ) =   15.00
                                      
P (   90   < X <    100   ) =    P (    7.5   < Z <    15.0   )   
                                      
= P ( Z <    15.00   ) - P ( Z <   7.50   ) =    1.00000   -    1.00000   =    0.00000000000003186

Please revert in case of any doubt.

Please upvote. Thanks in advance