This paper reviews definitions associated with probability and normal distributions along with solving critical value and binomial approximation problems. A key highlight of the normal distribution, was the use of confidence interval methods to estimate population parameters from sample proportions. Normal distribution curves were also used to illustrate the continuity correction.
¶ … Random Variable for Each Statement as Being Discreet or Continuous by
(a) the number of freshman in the required course, English 101
A) Discreet B) Continuous
(b) the number of phone calls between Florida and New York on Thanksgiving day.
A) Discreet B) Continuous
(c) the height of a radomly selected student.
A) Discreet B) Continuous
(d) the number of spills that occur in a local hospital.
A) Discreet B) Continuous
(e) the braking time of a car.
A) Discreet B) Continuous
Provide an appropriate response.
List the four requirements for a binomial distribution.
(i) Observations are independent
(ii) Outcome of observation is either a success or failure
(iii) Probability of outcome is the same
(iii) Fixed number of observations
Identify each of the variables in the binomial probability formula.
P (x) = __ n!__ . px . qn-x
(n -- x)! x!
n = number of trials x = number of successes p = probability of success q = probability of failure
Also, explain what the fraction __ n!____ computes.
(n -- x)!x!
Number of ways to select 'x' items from 'n' given items
4. Assume that a procedure yields a binomial distribution with a trial repeated n times. Use the binomial probability formula to find the probability of x successes given the probability p of success on a single trial.
n = 12, x = 5, p = 0.25, q = 0.75
P (x = 5) = __ 12!__ . (0.25)5 . (0.75)(12-5)
(12 -- 5)! 5!
= __ 12!__ . (0.25)5 . (0.75)7
7! 5!
= 12 x 11 x 10 x 9 x 8 x 7!_ . (0.000977). (0.133484)
7! x 5 x 4 x 3 x 2 x 1
= 0.103241
CHAPTER 6
1. The Precision Scientific Instrument Company manufactures thermometers that are supposed to give readings of 0°C at the freezing point of water. Test on a large sample of these instruments reveal that at the freezing point of water, some thermometers give reading below 0° (denoted by negative numbers ). Assume that the mean reading is 0°C and that standard deviation of the reading is 1.00°C. Also assume that the readings are normally distributed. If one thermometer is randomly selected the, find the probability that at the freezing point of water, the reading is less than 1.57°C.
Z-score
Area
1.5 + 0.07 = 1.57
0.9418
This is already standardized,
2. If Z. is the standard variable, find the probability, that Z. lies between 0.7 and 1.98
Z-score
Area
0.7 + 0.00 = 0.70
0.7580
1.9 + 0.08 = 1.98
0.9761
3. Use the continuity correction and describe the region of the normal curve that correspond to the indicated binomial probability. Make a diagram for each.
i) the probability of at least 150 passengers on your next commercial flight.
P (X ? 150) = P (X > (150-0.5)) = P (X > 149.5)
ii) the probability of no more than 6 absent students in a statistics class.
P (X ? 6) = P (X < (6 + 0.5)) = P (X < 6.5)
iii) the probability that fewer than 24 students understanding continuity correction.
P (X < 24) = P (X < (24-0.5)) = P (X < 23.5)
iv) the probability of exactly 46 marbles.
P (X = 46) = P ((46-0.5) < X < (46 + 0.5)) = P (45.5 < X < 46.5)
Chapter 7
DIRECTIONS: Match the description on the left with the correct word or name on the right. Write the correct letter in the space provided.
Question # 1.
i) ____E____ is the probability 1 - (
A. Margin of Error
ii) __F____ a range of values used to estimate the true B. upper confidence value of a population parameter. limit minus the lower
. confidence limit, divided . By 2.
iii) __B____ maximum error of the estimate C. q = 1 -- p
iv) ____G____ critical value D. Point Estimate
v) ____A__ EE. Confidence level
vi)____D____ is a single value used to approximate a F. Confidence interval population parameter.
vii)____C____ sample proportion of failures in a G. z (/2
sample of size n.
2. According to the text what are the 5 observations needed to determine critical values?
i) Sample proportions approximated by normal distribution
ii) level of significance (?)
iii) two-tailed (?/2) to identify rejection/critical region(s)
iv) Confidence level (1 - ?)
v) Determine critical value Z?/2 (from z score tables)
3. Find the critical value corresponding to a 98% of confidence. Use Table A2.
(Make a bell-shaped graph in answering this question)
Z-score
Area
2.05
0.9798
Critical value
0.98
2.06
0.9803
Interpolating gives:
4. Find the margin of error (E) for the 95% confidence interval used to estimate the population proportion.
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