Question

1. Statistics students believe that the mean score on a first statistics test is 65. The instructor thinks that the mean score is higher. She samples 10 statistics students and obtains the scores:

Grades	61.9	74.4	63.9	96	64.3	69	62.7	88	85.5	74.4

Test grades are believed to be normally distributed.

Use a significance level of 5%.

1. State the alternative hypothesis: HA:HA:
   • μ≠65μ≠65
   • μ>65μ>65
   • μ<65μ<65
2. State the mean of the sample: ____
3. State the standard error of the sample means: ______
4. State the test statistic: t= ______
5. State the p-value: ______
6. Decision:
   • Fail to reject the null hypothesis.
   • Reject the null hypothesis.

2. Nationally, patients who go to the emergency room wait an average of 5 hours to be admitted into the hospital. Do patients at rural hospitals have a lower waiting time? The 11 randomly selected patients who went to the emergency room at rural hospitals waited an average of 4.8 hours to be admitted into the hospital. The standard deviation for these 11 patients was 2.2 hours. What can be concluded at the the αα = 0.01 level of significance level of significance?

1. For this study, we should use Select an answer t-test for a population mean z-test for a population proportion
2. The null and alternative hypotheses would be:

H0:H0:  μ or p ;  ≠ or < or = or > ? _____   

H1:H1:  p or μ ; ≠ or = or < or > ? _______

3. The test statistic z or t? = ______ (please show your answer to 3 decimal places.)
4. The p-value = _____ (Please show your answer to 4 decimal places.)
5. The p-value is  > or ≤ ? α
6. Based on this, we should Select an answer, fail to reject or accept or reject  the null hypothesis.
7. Thus, the final conclusion is that ...
   • The data suggest the populaton mean is significantly lower than 5 at αα = 0.01, so there is statistically significant evidence to conclude that the population mean waiting time to be admitted into the hospital from the emergency room for patients at rural hospitals is lower than 5 hours.
   • The data suggest that the population mean awaiting time to be admitted into the hospital from the emergency room for patients at rural hospitals is not significantly lower than 5 hours at αα = 0.01, so there is statistically insignificant evidence to conclude that the population mean waiting time to be admitted into the hospital from the emergency room for patients at rural hospitals is lower than 5 hours.
   • The data suggest the population mean is not significantly lower than 5 at αα = 0.01, so there is statistically insignificant evidence to conclude that the population mean waiting time to be admitted into the hospital from the emergency room for patients at rural hospitals is equal to 5 hours.

Answer 1

1)

a)

Ha :   µ >   65       (Right tail test)  
                  

b)

Sample Mean,    x̅ = ΣX/n =    74.0100  

c)

sample std dev ,    s = √(Σ(X- x̅ )²/(n-1) ) =   12.0544
Sample Size ,   n =    10
Standard Error , SE = s/√n =   12.0544   / √    10   =   3.8119  

d)

t-test statistic= (x̅ - µ )/SE = (   74.010   -   65   ) /    3.8119   =   2.36

e)

p-Value   =   0.0212   [Excel formula =t.dist(t-stat,df) ]  

f)

Decision:   p-value<α, Reject null hypothesis   
---------------------------------

2)

a)  t-test for a population mean

b)

Ho :   µ =   5
Ha :   µ <   5

c)

sample std dev ,    s =    2.2000                  
Sample Size ,   n =    11                  
Sample Mean,    x̅ =   4.8000                  
                          
degree of freedom=   DF=n-1=   10                  
                          
Standard Error , SE = s/√n =   2.2000   / √    11   =   0.6633      
t-test statistic= (x̅ - µ )/SE = (   4.800   -   5   ) /    0.6633   =   -0.302

d)

p-Value   =   0.3846   [Excel formula =t.dist(t-stat,df) ]

e)

Decision:   p-value>α,

f)

fail to reject the null hypothesis.

g)The data suggest that the population mean awaiting time to be admitted into the hospital from the emergency room for patients at rural hospitals is not significantly lower than 5 hours at αα = 0.01, so there is statistically insignificant evidence to conclude that the population mean waiting time to be admitted into the hospital from the emergency room for patients at rural hospitals is lower than 5 hours.