Sample proportion questions ? (1 Viewer)

[stupidpersonlol]

Hi everyone,

Seeing how Advanced was so focused on new syllabus stuff I'm lowkey concerned that ext 1 is also gonna be like that. Does anyone have any sample proportion questions by any chance? I don't think I'm familiarised enough with it. All good if not. Thank youuuuu

 

[jimmysmith560]

Here are a few questions that I believe would help:

1. Suppose that 45% of all HSC students exercise at least 4 days each week. If a random sample of 50 students is taken, what is the probability that at least 80% of them exercise at least 4 days per week?

2. It is known that 24% of HSC students do not have a driver’s licence. In a random sample of 16 HSC students, what is the probability that half of them will not have a driver’s licence?

3. A computer simulation is designed to draw random samples of size from a large dataset. The proportion of the population that exhibits a certain characteristic is. If represents the sample proportion exhibiting the characteristic under investigation, find the largest sample size that should be used so that the standard deviation of is at least 0.01.

4. A manufacturer makes earbuds that have a probability of 0.02 of being defective. Quality control officers test random samples of 50 earbuds each hour and reject the earbuds made in that hour if at least 3 earbuds are defective. Find the probability that the earbuds made in any hour will be rejected. Answer to 2 significant figures.

5. It is estimated that approximately 45% of Australians will experience a mental health condition in their lifetime. If a random sample of 120 mature adults were surveyed, what is the probability of 48 or more having experienced a mental health condition?

I hope this helps!

 

[stupidpersonlol]

Here are a few questions that I believe would help:

Suppose that 45% of all HSC students exercise at least 4 days each week. If a random sample of 50 students is taken, what is the probability that at least 80% of them exercise at least 4 days per week?

It is known that 24% of HSC students do not have a driver’s licence. In a random sample of 16 HSC students, what is the probability that half of them will not have a driver’s licence?

A computer simulation is designed to draw random samples of size from a large dataset. The proportion of the population that exhibits a certain characteristic is. If represents the sample proportion exhibiting the characteristic under investigation, find the largest sample size that should be used so that the standard deviation of is at least 0.01.

A manufacturer makes earbuds that have a probability of 0.02 of being defective. Quality control officers test random samples of 50 earbuds each hour and reject the earbuds made in that hour if at least 3 earbuds are defective. Find the probability that the earbuds made in any hour will be rejected. Answer to 2 significant figures.

It is estimated that approximately 45% of Australians will experience a mental health condition in their lifetime. If a random sample of 120 mature adults were surveyed, what is the probability of 48 or more having experienced a mental health condition?

I hope this helps!

aww thank you jimmy

 

[jimmysmith560]

aww thank you jimmy

No worries, best of luck!

 

[walaili]

hi would you have the solutions to this?

 

[jimmysmith560]

hi would you have the solutions to this?

Yep! I will organise the solutions into spoilers, as follows:

Spoiler: Note

Note: For all questions, apply the tests and to determine if the distribution of the sample proportions can be approximated using the normal distribution.

Spoiler: Solution to Question 1

The distribution of the sample proportions can be approximated using the normal distribution with:



The probability that at least 80% of them exercise at least 4 days per week is approximately zero. Alternatively solve using binomial probability:

Spoiler: Solution to Question 2

The distribution of the sample proportions cannot be approximated using the normal distribution.

Solve using binomial probability:

Spoiler: Solution to Question 3

,

Assuming the distribution of the sample proportions can be approximated using the normal distribution then:

Solve for n such that



The largest sample size that can be used is 1875.

Spoiler: Solution to Question 4

The distribution of the sample proportions cannot be approximated using the normal distribution.

Solve using binomial probability:

The solution to Question 5 will be included in the next post.

 

[jimmysmith560]

Here is the solution to Question 5:

Spoiler: Solution to Question 5

The distribution of the sample proportions can be approximated using the normal distribution with:



48 represents a sample proportion of 0.4

I hope this helps!

 

[walaili]

Yep! I will organise the solutions into spoilers, as follows:

Spoiler: Note

Note: For all questions, apply the tests and to determine if the distribution of the sample proportions can be approximated using the normal distribution.

Spoiler: Solution to Question 1

View attachment 33845

The distribution of the sample proportions can be approximated using the normal distribution with:

View attachment 33846

The probability that at least 80% of them exercise at least 4 days per week is approximately zero. Alternatively solve using binomial probability:

View attachment 33847

Spoiler: Solution to Question 2

View attachment 33848

The distribution of the sample proportions cannot be approximated using the normal distribution.

Solve using binomial probability:

View attachment 33849

Spoiler: Solution to Question 3

,

Assuming the distribution of the sample proportions can be approximated using the normal distribution then:

Solve for n such that View attachment 33850

View attachment 33851

The largest sample size that can be used is 1875.

Spoiler: Solution to Question 4

View attachment 33852

The distribution of the sample proportions cannot be approximated using the normal distribution.

Solve using binomial probability:

View attachment 33853

The solution to Question 5 will be included in the next post.

hi thanks so much, i thought it was np>5 instead of 10 to apply the normal approximation? or did my teacher give me the wrong info

 

[Siwel]

hi thanks so much, i thought it was np>5 instead of 10 to apply the normal approximation? or did my teacher give me the wrong info

there are different ways to justify, i think as long as you say what one you are doing and justify it should be alright

 

[walaili]

thanks everyone! btw could someone please clarify what p(hat) means? (ok not 'fat')

like what would p(hat)=X/n be?

eg for Q1, is p(hat) X/n = 0.8/50? since X=0.8 or am I wrong?

 

[walaili]

and for Q2, can we use the continuity correction instead? or is it still too small

 

[Siwel]

and for Q2, can we use the continuity correction instead? or is it still too small

whats continuity correction?

 

[jimmysmith560]

thanks everyone! btw could someone please clarify what p(hat) means? (ok not 'fat')

Essentially, p-hat is the proportion of individuals in the sample who have that particular characteristic. In other words, it is the number of individuals in the sample who have that characteristic of interest divided by the total sample size (n).

 

[stupidpersonlol]

whats continuity correction?

i asked my teacher and she said its most likely not in the new syllabus

 

[Siwel]

i asked my teacher and she said its most likely not in the new syllabus

yeah never heard of continuity correction but the sample approx becoming normal distribution is in there i believe

 

[stupidpersonlol]

yeah never heard of continuity correction but the sample approx becoming normal distribution is in there i believe

yes defs...im so screwed for tmr lol

 

[Siwel]

yes defs...im so screwed for tmr lol

if its as much stats as adv im screwed

 

[stupidpersonlol]

if its as much stats as adv im screwed

i feel like it is ;(

 

[Jerryclover]

Note: For all questions, apply the tests

and

to determine if the distribution of the sample proportions can be approximated using the normal distribution.

Wait... Is this necessary? I know this exists but have never seen it used in answers to questions on sample proportions

 

[Trebla]

Wait... Is this necessary? I know this exists but have never seen it used in answers to questions on sample proportions

Those thresholds are a “guide”, not a hard and fast rule. Very unlikely you will get questions that are right on the margins of that threshold. Most of the time it will be pretty obvious whether a normal approximation is required or not. At the end of the day, the approximation exists to compensate for an overly cumbersome calculation.