Create a probability distribution table

Assignment Help Applied Statistics

Reference no: EM132263830

Statistics Questions -

Q1. a) Create a probability distribution table using the following data of exam scores of 20 students in a class. The probability distribution table is the list of possible output values and corresponding probabilities (or percentages).

10, 15, 10, 20, 25, 10, 20, 30, 35, 15, 20, 10, 35, 25, 20, 20, 10, 25, 20, 25

(Hint: The probability distribution table should look like the following (This is just an example, you need to create similar table from the data above).

No of Cars (X)	Probability (%)
30	0.1
35	0.2
40	0.4
45	0.2
50	0.1

b) What is the total probability? What does this mean? Can probability be less than 0; equal to 0; and more than 1. Justify with reasons.

c) Let X be the exam score, use the above information to evaluate the followings

(i) P(X=25)

(ii) P(X<25)

(iii) P(X≥20)

Q2. Categorize each of the followings as continuous or discrete. The random variables are:

(a) A is 'the age in completed years of the first person I see wearing a hat'.

(b) B is 'the length of the next car to enter the parking lot'.

(d) D is 'the date next July of the day with the highest temperature'.

Q3. Let X= Sum of scores when two dice are rolled (e.g. if you roll two dice and obtained 1 in first and 3 in second dice, then X is 4). Create a probability distribution table of X.

Q4. Write down the three main properties of normal distribution?

Q5. Use empirical rule (68-95-99.7% rule) to calculate the following:

SAT (combined) scores of 200 college-bound seniors in high school has the normal distribution with mean 1050 and standard deviation 150.

a) What percentage of seniors scored between 900 and 1200?

b) What percentage of seniors scored more than 750?

c) What percentage of seniors scored between 600 to 1200?

d) What percentage of seniors scored more than 1050?

e) What is the 84th percentile?

Note: n^th percentile is the value for which n% are below that value. So 84th percentile indicates the value for which 84% of seniors will get less than that score.

f) Find the value x such that the 2.5% of all seniors have SAT score below x.

g) Find the value x such that the 16% of all seniors have SAT score below x.

h) What are the 2.5th and 97.5th percentiles of this distribution?

i) How many seniors scored less than 1200?

j) How many seniors scored between 600 and 1350?

Q6. As you increase the sample size of a random sample,

(i) What happens to the standard deviation of the sample mean? Does it increase, decrease or remains the same?

(ii) What happens to the sample mean, will it be closer to the true population mean (more accurate), or remains the same or will be less accurate?

(iii) What happens to the distribution of sample mean, becomes wider and skewed, or remains same, or approaches to the normal distribution?

Q7. State central limit theorem (CLT).

Q8. It is assumed that height of teenage students in a population is normally distributed with mean=52" and standard deviation =3". A simple random sample of 30 teenage students is taken and sample mean is calculated. If several such samples of same size are taken

(i) What could be the mean of all sample means?

(ii) What could be the standard deviation of all sample means?

(iii) Will the distribution of sample means be normal?

(iv) Write down the distribution of sample mean in the form of x^-~ N ( , /√n).

Q9. A packaging plant fills bag with cement. The weight of cement X is normally distributed with mean 25 lbs. and standard deviation 4 lbs. Use this information to standardize the following weights.

(i) 30 lbs.

(ii) 25 lbs.

(iii) 23 lbs.

Q10. A packaging plant fills bag with cement. The weight of cement X is normally distributed with mean 25 lbs. and variance 25 lbs. Three samples of size 30 are drawn and their sample means are recorded (below). Standardize the following sample means.

(i) 24 lbs.

(ii) 26 lbs.

(iii) 27 lbs.

Q11. It is assumed that height of teenage students in a population is normally distributed with mean=52" ("indicates inches). Two simple random samples of 15 teenage students were taken and sample means and sample variances were calculated. The means for first and second samples were 50" and 54"; and sample variances for first and second samples were 16" and 25" respectively. Standardize the sample mean values of both samples.

Q12. Use the following R command to create norm_data

norm_data<-rnorm(1000,mean=10,sd=4)

(i) Create a histogram of norm_data. Comment on the shape of the histogram.

(ii) What % of values are (use pnorm() function in R)

a. less than 8

b. more than 13

c. between 7 and 12

d. less than 10?

(iii) What are the 75th and 45th percentiles of this data? (Use qnorm() function).

(iv) If 95% values are less than K, what is K?

Reference no: EM132263830

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