### Test 02 | 7.1 to 9.2 • Name:

##### Chapter Seven: Normal curve problems
NameEssay score x/40
Saturnina30
Jehosephat35
Theophillia32
Cuteleen29
Fashionette36
Jamesbond19
Santa20
Rambo29

The data is eight actual student essay scores from PICS high school, the names are fictitious.

1. x = ______________ Calculate the sample mean.
2. sx = ______________ Calculate the sample standard deviation.

Presume that the essay scores were normally distributed. The following problems derive from chapter seven. Use the standard normal distribution to determine your answers. Use the sample mean x from question one for µ, and the sample standard deviation sx from question two for σ in the questions below. Use Excel to calculate z values and probabilities as appropriate. Here the variable is x and NOT x.

1. p(x ≥ 28.75) = ______________ Based on the eight student sample above, what percentage of all the student essays from PICS scored 28.75 or more?
2. p(x ≥ 32) = ______________ Thirty-two is considered a strong score on the essay. What percentage of student essays scored 32 or higher?
3. ______________ 421 students took the test at PICS this past March. Use the percentage above to calculate how many students scored 32 or higher.
4. p(x ≤ ______________) = 0.25 What is the essay score below which the lowest 25% of the students scored?
##### Chapter nine: Confidence interval calculations

Use the sample mean x from question one and the sample standard deviation sx from question two and the data given in the first table to calculate a 95% confidence interval for the population mean µ essay score using the student's t-distribution.

1. sample size: n = ______________
2. confidence level: c = ______________
3. degrees of freedom: = ______________
4. t-critical: tc = ______________
5. Error tolerance E: = ______________
6. Calculate the confidence interval for the population mean essay score based on the eight essay sample:

P( ___________ ≤ µ ≤ ___________ ) = 0.95
7. _________ The actual population mean student essay score for PICS this year is 25.56. Does the 95% confidence interval for the eight essay sample include 25.56?
8. _________ The actual population mean student essay score for PICS in spring 2005 was 22.63. Does the 95% confidence interval for the eight essay sample include 22.63?

t-critical values for c = 0.95
d.f.tc
42.7764
52.5706
62.4469
72.3646
82.3060
92.2622
102.2281
zc1.9600
Statistic or Parameter Symbol Equations Excel
Chapter seven: Normal statistics
Calculate a z value from an x z = =STANDARDIZE(x, µ, σ)
Calculate an x value from a z x = z σ + µ =z*σ+µ
Find a probability p from a z value     =NORMSDIST(z)
Find a z value from a probability p     =NORMSINV(p)
Chapter eight: Distribution of the sample mean x
Calculate a z-statistic from an x z =(x - µ)/(sx/SQRT(n))
Calculate a t-statistic (t-stat) t =(x - µ)/(sx/SQRT(n))
Calculate an x from a z   =µ + zc*sx/sqrt(n)
Chapter nine: Confidence interval statistics
Find a tc value from a confidence level c and sample size n tc   =TINV(1-c,n-1)
Calculate an error tolerance E of a mean for any n ≥ 5 using sx. E =tc*sx/SQRT(n)
Calculate a confidence interval for a population mean µ from a sample mean x and an error tolerance E   x-E≤ µ ≤x+E