Lesson test: The normal curve

1.[2p]
What fixes the constant $1 / (σ \sqrt{2 π})$ in front of the normal density?

The requirement that the total area under the curve be exactly $1$

Correct

The requirement that the mean be $μ$

The answer is: The requirement that the total area under the curve be exactly $1$

The requirement that the density never exceed $1$

The answer is: The requirement that the total area under the curve be exactly $1$

Nothing: it is chosen for convenience

The answer is: The requirement that the total area under the curve be exactly $1$

2.[2p]
What proportion of a normal distribution lies within two standard deviations of the mean? Give the answer to four decimal places.

CorrectNot quite: 0.9545

3.[2p]
Heights are normal with mean $175$ cm and standard deviation $7$ cm. What is the $z$ score of a height of $190$ cm? Give the answer to four decimal places.

CorrectNot quite: 2.1429

4.[3p]
For the same population, what proportion are taller than $190$ cm? Give the answer to four decimal places.

CorrectNot quite: 0.0161

5.[2p]
Two independent normal variables are added. What is the distribution of the sum?

Normal, with the means added and the variances added

Correct

Normal, with the means added and the standard deviations added

The answer is: Normal, with the means added and the variances added

Not normal in general

The answer is: Normal, with the means added and the variances added

Normal only if the two have the same variance

The answer is: Normal, with the means added and the variances added

6.[3p]
Four components, each normal with standard deviation $0.3$ mm, are stacked independently. What is the standard deviation of the total length, in mm?

CorrectNot quite: 0.6

7.[3p]
A fair coin is tossed $100$ times. Using the normal approximation with continuity correction, what is $P (X \geq 60)$ ? Give the answer to four decimal places.

CorrectNot quite: 0.0287

8.[3p]
Why does the continuity correction matter?

Without it the boundary is off by half a unit, which for $P (X \geq 60)$ changes the answer by about twenty percent

Correct

It corrects for the binomial not being independent

The answer is: Without it the boundary is off by half a unit, which for $P(X \ge 60)$ changes the answer by about twenty percent

It compensates for a small sample size

The answer is: Without it the boundary is off by half a unit, which for $P(X \ge 60)$ changes the answer by about twenty percent

It converts the variance from $n p$ to $n p (1 - p)$

The answer is: Without it the boundary is off by half a unit, which for $P(X \ge 60)$ changes the answer by about twenty percent

9.[3p]
Which of these are reasons the normal can be the wrong model?

Select all that apply

It assigns positive probability to values that are physically impossible

Correct

Its tails are far thinner than those of many real quantities

Correct

Quantities built by multiplication are log-normal rather than normal

Correct

Its mean and standard deviation cannot be estimated from data

The answer is: It assigns positive probability to values that are physically impossible, Its tails are far thinner than those of many real quantities, Quantities built by multiplication are log-normal rather than normal

1.[2p] What fixes the constant 1/(σ2π) in front of the normal density?

2.[2p] What proportion of a normal distribution lies within two standard deviations of the mean? Give the answer to four decimal places.

3.[2p] Heights are normal with mean 175 cm and standard deviation 7 cm. What is the z score of a height of 190 cm? Give the answer to four decimal places.

4.[3p] For the same population, what proportion are taller than 190 cm? Give the answer to four decimal places.

5.[2p] Two independent normal variables are added. What is the distribution of the sum?

6.[3p] Four components, each normal with standard deviation 0.3 mm, are stacked independently. What is the standard deviation of the total length, in mm?

7.[3p] A fair coin is tossed 100 times. Using the normal approximation with continuity correction, what is P(X≥60)? Give the answer to four decimal places.

8.[3p] Why does the continuity correction matter?

9.[3p] Which of these are reasons the normal can be the wrong model?

1.[2p]
What fixes the constant $1 / (σ \sqrt{2 π})$ in front of the normal density?

2.[2p]
What proportion of a normal distribution lies within two standard deviations of the mean? Give the answer to four decimal places.

3.[2p]
Heights are normal with mean $175$ cm and standard deviation $7$ cm. What is the $z$ score of a height of $190$ cm? Give the answer to four decimal places.

4.[3p]
For the same population, what proportion are taller than $190$ cm? Give the answer to four decimal places.

5.[2p]
Two independent normal variables are added. What is the distribution of the sum?

6.[3p]
Four components, each normal with standard deviation $0.3$ mm, are stacked independently. What is the standard deviation of the total length, in mm?

7.[3p]
A fair coin is tossed $100$ times. Using the normal approximation with continuity correction, what is $P (X \geq 60)$ ? Give the answer to four decimal places.

8.[3p]
Why does the continuity correction matter?

9.[3p]
Which of these are reasons the normal can be the wrong model?