Question

Compute a z score to determine the percentage chance of observing a country with 6 percent unemployment or more.

Answer 1

a) Draw the normal distribution here. Label information as you go to keep track of everything. Include the X and Z scales.

Answer 2

oof ight give me a sec

Answer 3

so u need the formula or nah

Answer 4

Honestly, I don't know.

Answer 5

oh well i dont have statistics till nex semester sooo

Answer 6

but thats a mood lmao

Answer 7

Thanks for trying.

Answer 8

well i didnt do anything tho but say idk tho

Answer 9

but ur welcome

Answer 10

/:

Answer 11

i do appoligize tho

Answer 12

First calculate the mean percent. Then find the absolute value difference of each x value from the mean. Then square each, sum them. Then divide that that sum by n, or the number of x values.

This gives you the standard deviation, which you can then input into this formula for the z-score

\[z = \frac{ x_{i} - \mu}{ \sigma } \]

Z score of 6% is equal to 6% - the mean, divided by the standard deviation

Formula for mean:
\[\mu = \frac{ \sum x }{ n }\]
Formula for standard deviation:
\[\sigma = \frac{ \sum D^2 }{ n }\]

Answer 13

Technically finding the absolute value difference doesn't matter (it's just how it's sometimes talked about when learning the concept). Just square each difference then sum them. That's sum D^2

Answer 14

Then once you calculate the z score, look at a z chart and find the z score. You're looking for the % that relates to the z-score.

Answer 15

What the actual fück. Shadow, I don't understand jack about what you just said. I'm a senior in college majoring in Political Science and I haven't done math in 2 years. I'm losing my mind - just answer the question and stop playing with me.

Answer 16

lol

Answer 17

Please, I've learned enough. Just answer the question. It's 3 AM and I want to sleep.

Answer 18

Is the Z-score 6 meaning it is Luxembourg or Ireland? Or both?

Answer 19

hmm, I couldn't tell you

Answer 20

SHADOW PLEASE

Answer 21

Oh I see what you mean now. 6% in the questions wording is just a random # they chose

Answer 22

What? Why would they do that? Btw, the class isn't Statistics. It's Research Analysis in Political Science.

Answer 23

I have my own stuff to do. This stuff is all algebra. The mean is the average.

1, 2, 3

Average = (1 + 2 + 3)/3

You can do this entire problem in a calculator / Wolfram.

Answer 24

\[\sum \] just means sum. When it's next to x, it means add up all the x values as shown in the table.

Answer 25

What are X and Z scales?

Answer 26

Why would they mention 6 if it didn't matter?

Answer 27

How do you find the normal distribution? (I've already asked Google).

Answer 28

That's a normal distribution

Answer 29

Half the scores on one side, other half on the others

Answer 30

Average is in the center.

Along the x -axis are the z scores which relate to percents.

Answer 31

You're looking for the z score that relates to 6% and the mean of this distribution along with the distributions standard deviation.

Answer 32

Once you calculate the z value you know if it's negative or not

Answer 33

Okay, the question says "compute a z score to determine the percentage chance of observing a country with *6 percent* unemployment or more." Does that mean we are not counting Monaco, United Kingdom, Netherlands?

Answer 34

The average means you sum (add up) all the x values, that means the scores of all the countries

Answer 35

Because one of the questions is "What is the percentage chance of observing a country with 6 percent unemployment or more?"

Answer 36

To get the average unemployment rate across those countries

Answer 37

Yeah is this due today?

Answer 38

No but I need to finish this right now.

Answer 39

Statistics at 3am, you're definitely weird

Answer 40

Please leave me alone. You don't understand.

Answer 41

Do you know anyone that can actually help me?

Answer 42

\[Average = \mu = \frac{ 7 + 10 + 6 + 6 + 2 + 5 + 4 }{ 7 } = \frac{ 40 }{ 7 } = 5.71\]

Answer 43

I know it's 3am for you but play nice (:

Answer 44

Next you basically take each x value and subtract the mean from it. This gives you the difference, or, how much it deviates away from the average.

Answer 45

Nice? I'm livid right now. Imagine spending hours on this stupid asś question and still not understanding. I am a POLI SCI major for a reason.

Answer 46

The x values are the countries, right?

Answer 47

Yes

Answer 48

Once you get the differences you square each of them -> x^2, then add them all up. This gives you \[\sum D^2\]
Then you calculate the standard deviation using the formula I provided which is just basic inputs of sum D^2 /n

Answer 49

One small correction the formula should be:
\[\sigma ^2 = \frac{ \sum D^2 }{ n }\]

Answer 50

Is this starting to make more sense / look doable

Answer 51

So are we calculating the values less that 6 or not? Yes, it's starting to look doable. I'm subtracting right now.

Answer 52

We are. I see what you're thinking but I think it might make more sense once we get to the z score.

Answer 53

Okay, I subtracted and I got the following: 1.29, 4.29, 0.29, 0.29, -3.71, -0.71, and -1.71.

Answer 54

I would keep them in fraction form. In cases like this decimal points matter and the fraction encompasses all the possible decimals.

Answer 55

It's a lot of grunt work, I know

Answer 56

Fraction form? How do I find the fraction?

Answer 57

So I would do 7 - (40/7) in a calculator then use the fraction form of the result.

Answer 58

It's just to play it safe.

Answer 59

Where did you get 40 from?

Answer 60

Wait, never mind.

Answer 61

I'm not getting fractions. I'm getting mixed fractions.

Answer 62

Is that bad?

Answer 63

no, I'm getting them too

Answer 64

Okay, thank you.

Answer 65

Okay, I got 1 2/7, 4 2/7, 2/7, 2/7, -3 5/7, -5/7, and -1 5/7.

Answer 66

Now, I'll be squaring those 7 values by 2, correct?

Answer 67

Yes

Answer 68

I got 4/49 for the first one. Is that correct? I want to make sure before I do the rest.

Answer 69

The way I calculated my differences is x - 40/7

I got 9/7, 30/7, 2/7, 2/7, -26/7, -5/7, -12/7

Squared: 81/49, 900/49, 4/49, 4/49, 676/49, 25/49, 144/49

Summed: 262/7

Answer 70

I used an online calculator because I couldn't find mine. Are you telling me it's wrong?

Answer 71

Take 4 2/7

4 is basically 4 7/7s, so 4 x 7 = 28

28 + 2 = 30

30/7

Answer 72

Same stuff

Answer 73

Okay, cool but whose answers should we go with? Probably yours.

Answer 74

Lol

Answer 75

\[\sigma^2 = \frac{ \frac{ 262 }{ 7 } }{ 7 } = \frac{ 262 }{ 49 }\]
\[\sigma = \sqrt ( \frac{ 262 }{ 49 })\]

Answer 76

Then we just input that into our z formula

Answer 77

\[z = \frac{ x - \mu }{ \sigma } = \frac{ 6 - \frac{ 40 }{ 7 } }{ \sqrt \frac{ 262 }{ 49 } }\]

Answer 78

I thought the number 6 in the question didn't matter?

Answer 79

I just meant that they could've chosen any %

Answer 80

By the way your prof sucks. These numbers are hella ugly

Answer 81

I got sqrt 2/131 or 0.12356

We find that on a z chart

Answer 82

I know he sucks. It's an online class so there are zero resources or lectures. It's horrendous. We're basically teaching ourselves this sh!t.

Answer 83

Wait seriously, no lectures?

Answer 84

The hell

Answer 85

Idk how the others in class are doing it.

Answer 86

Where do we find a z chart?

Answer 87

k I referred to a z chart and for 0.12 I got 0.54776 is to the left of the z score. Take 1 - 0.54776 and you have the % of scores in this distribution that are to the right of 6%

Answer 88

I used this one https://www.math.arizona.edu/~rsims/ma464/standardnormaltable.pdf

Answer 89

The column is the first decimal point, then the row is the second decimal point.

Answer 90

See how 'more than 6%' kind of makes sense now?

Answer 91

Um, I can't find 0.12.

Answer 92

Look for 0.1 under z, then use the row to find 0.02, then find the intersection of that row and column value.

Answer 93

I've had this concept taught to me in two different classes in lecture, and I can't imagine having to learn this on your own lmao

Answer 94

Okay, I got 0.45224. What do I do with this value?

Answer 95

That's the %

Answer 96

Of scores in the distribution that are above that z value

Answer 97

Or % of countries that have an unemployment rate of 6% or higher

Answer 98

We needed all the countries including the ones below 6% to get a 'feel' for the distribution, to see how the scores were deviating. Then the formula is able to pinpoint what kind of z score 6% has in relation to the standard deviation and the mean.

Answer 99

Okay. Thank you. To answer the question, do I have to draw an entire graph? Can I just say the normal distribution is 0.54776 and list the x and z scales.

Answer 100

I think sometimes in class, we'd draw the distribution, mark the z value, then shade in the %

Answer 101

Basically just another was of saying 'the % is greater than z = 0.12'

Answer 102

Also remember 0.45224 is the % you want to list

Answer 103

or 45.2%

Answer 104

54.8% is behind the z score

Answer 105

What?

Answer 106

But it's only asking for the normal distribution.

Answer 107

45.2% of the countries in this distribution are above 6%

Answer 108

\(\color{#0cbb34}{\text{Originally Posted by}}\) @Shadow
Created with RaphaëlReply Using Drawing
\(\color{#0cbb34}{\text{End of Quote}}\)
Okay, how about I attach your wonderful graph, list the x and z scales and include the normal distribution. That is more than what this professor deserves. Does that answer the question?

Answer 109

Sure, but what do you mean by 'normal distribution'

Answer 110

The question says "draw the normal distribution here" so I'll say the normal distribution is 0.54776.

Answer 111

Yeah but that makes no sense

Answer 112

The graph I drew is the normal distribution

Answer 113

But how will he know the normal distribution? Like the number?

Answer 114

54.8% is just the stuff left of the z score, not the normal distribution

Answer 115

Okay, genius, what do you think I should answer then?

Answer 116

Just attach your graph and mention the x and z scales? Will that answer the question?

Answer 117

The graph technically has the z value and the % which he's asking for, it's just kind of Evil hand-drawing. I'm not sure how he wants the graph represented but it's definitely readable in terms saying what the distribution of this question is.

Answer 118

He deserves an evil hand-drawn graph. I hope he goes blind reading it.

Answer 119

I'm used to answering stat questions differently but honestly hope all your uni classes aren't like this.

Answer 120

Are you full online or something?

Answer 121

They're not. This is the only awful class. It's a Poli Sci class so I don't think I need to be over-emphasize.

Answer 122

All my classes are offered online so I didn't have a choice.

Answer 123

Didn't know Poli-Sci did stats. Almost majored in that too xD

Answer 124

They don't! That's why I'm losing my mind. This is the first Poli Sci class that's putting me through this nightmare.

Answer 125

And I'm taking all online too. I just mean all of them provide actual lectures to watch. Having no resource but a textbook means you just paid for a textbook and hw. Not a professor.

Answer 126

Don't even get me started on how we're being ripped off.

Answer 127

Lmao, okay. I'd pick your brain about poly sci but it's late for you.

Answer 128

We don't even have classes where he explains the stuff.

Answer 129

Sir, I have more questions to answer after this. You honestly think I'm done?

Answer 130

Cause I just can't imagine a QC/OS user doing poly sci

Answer 131

Wait, why?

Answer 132

Think they'd all be business majors or other mainstream stuff. Not care about politics.

Answer 133

This is it.

Answer 134

It's not just about politics but yeah, that's a major component.

Answer 135

What is 5.71

Answer 136

The z value? AKA the mean?

Answer 137

Does it have to be in fraction form again?

Answer 138

z value isn't the mean

Answer 139

Yes, it is? Didn't we subtract the x and z values?

Answer 140

Think he wants you to draw in the numbers below the x axis, as the z numbers.

Answer 141

Otherwise, idk what it's asking for

Answer 142

If the standard deviation is 5.71, then what is the mean?

Answer 143

Actually I had that right. Write the z values -1, 1, etc on the graph. Then under each write the appropriate SD. So for 1 its 5.71 and 2 it's 2 times 5.71 = ? etc

Answer 144

It's all coming back to me now lol

Answer 145

Dude, where are you getting the z values from?

Answer 146

Why did we subtract x and z (5.71) values if that's not even the z value?

Answer 147

5.71 is the average, not a z value

Answer 148

A z value of 5.71 would be in far right aka Africa

Answer 149

idk where you got that from but just write the z values as is shown in the image I lined, -3, -2, -1, 0, 1, 2, 3, then under them the appropriate standard deviations

Answer 150

Think that's what he's asking for.

Answer 151

Lol

Answer 152

Okay, I just scrolled up and I think I'm hallucinating. You never said 5.71 was the z value. It's the mean.

Answer 153

Yeah I'm starting to hallucinate too at this point

Answer 154

Africa sounds nice, maybe i'll die there and won't have to do this

Answer 155

lol

Answer 156

I genuinely laughed. I feel your pain. Unfortunately though we have to wrap this question up. I have my last minute online class procrastination stuff to do as well.

Answer 157

I can help you more tomorrow

Answer 158

Glad you got a laugh out of my misery. Thank you for bearing with me. I appreciate it. I'll try to answer the other questions and send you a screenshot to make certain they are correct. Thank you, legend :D

Answer 159

You're welcome. Good luck with your classes.

Answer 160

Answered the questions I could. Let me know if they're correct.

Answer 161

\(\color{#0cbb34}{\text{Originally Posted by}}\) @Shadow
\[\sigma^2 = \frac{ \frac{ 262 }{ 7 } }{ 7 } = \frac{ 262 }{ 49 }\]
\[\sigma = \sqrt ( \frac{ 262 }{ 49 })\]
\(\color{#0cbb34}{\text{End of Quote}}\)

Sigma is the standard deviation. This is my bad but what you did you 5.71 under the z scores should have been done to the SD. This is usually why I don't help late at night when my brain is dead xD My apologies.

But that should allow you to put in the decimal form of sigma as SD or standard deviation, then e) is what we discussed before, 45.2%

Answer 162

No idea what you just said.

Answer 163

Lol

Answer 164

Do we know the population SD? There's a difference between a standard SD and a pop SD.

Answer 165

hmm you actually making sense ._.

Answer 166

-_-

Answer 167

I think that's what we did find though

Answer 168

"Population standard deviation looks at the square root of the variance of the set of numbers."

Answer 169

Why? It didn't specify to find the population SD.

Answer 170

\[\sigma ^2 = variance \rightarrow \sigma = SD\]

Answer 171

and \[\sigma = sigm@\]

Answer 172

You're not making sense again.

Answer 173

Which answers of mine are incorrect?

Answer 174

\[\sigma = \sqrt \frac{ 262 }{ 49 } = 2.31\]

For z -3, -2, -1, 0, 1, 2, 3, the number under them should be this 2.31, so multiply 2.31 by -1, 1, 2, 3, etc. We did it for 5.71 but that was an error, as that was the mean.

Answer 175

If that helps you visualize it, the z markers are under the x axis, and one standard deviation lies on z = 1, so we put 2.31 under it.

Answer 176

\(\color{#0cbb34}{\text{Originally Posted by}}\) @Shadow
Actually I had that right. Write the z values -1, 1, etc on the graph. Then under each write the appropriate SD. So for 1 its 5.71 and 2 it's 2 times 5.71 = ? etc
\(\color{#0cbb34}{\text{End of Quote}}\)
????

Answer 177

Yeah I was hallucinating alongside you remember

Answer 178

Had some good sleep last night, now I see clearly now (:

Answer 179

Where did you get 2.31 from? Also, what is the mean? I can't find it.

Answer 180

The mean is the average, aka 5.71, derived from:
\(\color{#0cbb34}{\text{Originally Posted by}}\) @Shadow
\[Average = \mu = \frac{ 7 + 10 + 6 + 6 + 2 + 5 + 4 }{ 7 } = \frac{ 40 }{ 7 } = 5.71\]
\(\color{#0cbb34}{\text{End of Quote}}\)

Answer 181

I thought that was the SD? Is it the same?

Answer 182

\[\sigma = \frac{ \sum D^2 }{ n }\]
\[\sigma^2 = \frac{ \frac{ 262 }{ 7 } }{ 7 } = \frac{ 262 }{ 49 }\]
\[\sigma = \sqrt ( \frac{ 262 }{ 49 })\]
\[\sigma = 2.31\]

Answer 183

Remember when we added up the squared differences?

Answer 184

That's how we got to sum D^2, then to the variance sigma^2, then to sigma, which is the standard deviation

Answer 185

Please stop confusing me. Why do we have two SD's now?

Answer 186

We don't. The only SD is 2.31

Answer 187

I just simplified it from its previous forms

Answer 188

5.71 is the mean aka the average

Answer 189

No

Answer 190

Okay, got it.

Answer 191

Were c and d correct?

Answer 192

Yes

Answer 193

I gotta go take this test, will be back in like 2hrs if you have more stats stuff.

Answer 194

Updated answers. Not sure about e.

Answer 195

What you wrote just below appropriate SD is what you put for e

Answer 196

You can just put 45.2% though for e

Answer 197

\(\color{#0cbb34}{\text{Originally Posted by}}\) @candycove
Okay, I got 0.45224. What do I do with this value?
\(\color{#0cbb34}{\text{End of Quote}}\)

Unless he wants it rounded to the second place, which is 45.22% but it's whatever his standard is.

Answer 198

Ah, got it. Thank you so much. Don't you have a test?

Answer 199

Does this question fall under Statistics?

Answer 200

It does but I forget how to do this one. Real unfortunate because the process is rather fun.

Answer 201

The gist of it is that you'd calculate a z score and if it's over z = +- 1.96 then it would fall in the last 5% on either end of the distribution, which means that you could reject the null hypothesis (h0 says that the hypothesis isn't true), so when you reject it, you're saying your hypothesis is correct. 5% is a number chosen in science because it's basically saying 'it's highly likely that we're solid with this hypothesis.'

Answer 202

I used plus or minus 1.96 as this problem is asking for a two tailed one

Answer 203

Okay here's clean definitions from my textbook:

The null hypothesis (H0) basically says that there is no change, difference, or no relationship.

The alternative hypothesis (H1) states that there is a change, a difference, or a relationship for the general population.

Answer 204

So for this question H0 and H1 would be:

\[H_{0}: \mu = 23%\]
\[H_{1}: \mu \neq 23% \]

Answer 205

So since this is two tailed, if we get a z score over either -1.96 or 1.96, we can reject H0 since it's statistically significant that it isn't 23%, and we accept the alternative hypothesis.

Answer 206

\[z = \frac{ M - \mu }{ \sigma_{M} }\]

Where,
m = the sample mean
mu/mean/the average = the hypothesized population mean
sigma_{m} = standard error between M and mu

Answer 207

We will use this to calculate the z score

Answer 208

\[\sigma_m = \frac{ \sigma }{ \sqrt n }\]

Answer 209

So for the standard error you need to calculate the standard deviation as we did in the first problem, then divided it by the sqrt of the n values, so n = 7 here

Answer 210

\[z = \frac{ M - \mu }{ \sigma_{M} }\]

Where,
m = the sample mean
mu/mean/the average = the hypothesized population mean
sigma_{m} = standard error between M and mu

For this part, we're basically calculate the sample mean/average of the African countries, then subtracting 23% from it, then dividing it by the standard error which I explained above.

Find the standard deviation again will be a long trip math wise if you remember, but this is all doable stuff. Hopefully I've listed out all the steps clearly. I think you can do it.

Answer 211

It looks intimidating but stuff like "use a two-tailed test" is literally just saying 'we care if it's negative or positive number.'

Answer 212

For the z score

Answer 213

Feeling: stupid.

Answer 214

The question is asking for a sample confidence interval so why are we calculating the z score?

Answer 215

I wouldn't worry about it. You haven't learned this concept and it's the kind of concept that you need to have done a couple times to be comfortable with.

Answer 216

Are we looking for the sample SD or population SD?

Answer 217

The standard deviation of this sample

Answer 218

So we get sample mean, sum D^2, then SD of the sample. All the same stuff we did yesterday at 3am

Answer 219

Jesus

Answer 220

Stop using big words. Pick one: Sample SD or Population SD.

Answer 221

First lol

Answer 222

That reminds me, for one of those you have to do n-1...

Answer 223

First one in the picture or my text?

Answer 224

MY GOD I HATE MY LIFE

Answer 225

LOL

Answer 226

WHAT IS N-1

Answer 227

Wait, no. I don't want to know.

Answer 228

Stop laughing at me ,

Answer 229

Okay, so population?

Answer 230

The stuff on the right is basically sum D^2, thing on the left is whether you divide by n or n-1

Answer 231

We're using sample. Population is only when you're calculating the entire population which is rarely possible.

Answer 232

It's like if you had all the Charizard pokemon cards and H0 is not damaged and H1 is damaged

Answer 233

Kind of irrelevant but I had to remember it for my sake.

Answer 234

But we calculated the pop SD in the last question so it's not that rare.

Answer 235

SD: 19.19?

Answer 236

σ: 17.77?

Answer 237

That was fast.

Answer 238

I used a calculator. Study smarter not harder.

Answer 239

Which one is it?

Answer 240

Lol. If you used a calculator then you're probably right

Answer 241

or first

Answer 242

This is what I did.

Answer 243

lol

Answer 244

I mean it's just grunt work. You're really just showing you know what to do with the #s

Answer 245

The what?

Answer 246

Why are you mentioning new concepts to me. It's annoying.

Answer 247

I think we use sample SD formula here, because they specifically say "a sample of African countries'

Answer 248

Okay, so s: 19.19 -- what's next?

Answer 249

standard error, then make sure you have the sample mean to, we're calculating the z score after.

Answer 250

Standard Error
\[\sigma_m = \frac{ \sigma }{ \sqrt n }\]

Answer 251

7.25624291383

Answer 252

Guess you used the fraction since I got 7.253 and I just used 19.19

Answer 253

I used a calculator.

Answer 254

But now you just do (sample mean - 23%)/standard error

Answer 255

For the z score

Answer 256

I don't know how to do it otherwise. :)

Answer 257

I mean I just inputted (19.19)/(sqrt7). Usually I like to keep the entire decimal or the fraction since sometimes 1.95 vs 1.96 can be the difference between rejecting a hypothesis or not.

Answer 258

Is the sample mean 29.71?

Answer 259

I got the same

Answer 260

0.925?

Answer 261

nvm

Answer 262

k so this means it didn't change statistically significantly and we reject our alternative hypothesis

Answer 263

That imagine should express how it works to you, from my textbook.

Answer 264

Did you get the same z score? I'm not sure if mine is correct.

Answer 265

I got the same.

Answer 266

So it would fall in the in the middle 90% of the graph and not in the 5% tips of the distribution

Answer 267

Actually im dumb, it's 5% because in a two tailed, it's 2.5% each, in a one tailed its 5% at the end.

Answer 268

Okay but the question is asking for a confidence interval. Do we have one?

Answer 269

The confidence interval is alpla = 0.05, or the 5%, so it's the shaded region in the picture I uploaded.

Answer 270

alpha*

Answer 271

M = 5, 95% CI [4.82, 5.18] -- isn't that how a confidence interval is supposed to look like or does it not matter?

Answer 272

We follow this formula: https://www.mathsisfun.com/data/confidence-interval.html

Answer 273

\[\mu \pm z \frac{ \sigma }{ \sqrt n}\]

Answer 274

or just

\[\mu \pm z \sigma_{m}\]

Answer 275

Okay, what does that look like for this question?

Answer 276

sample mean plus or minus z score times the standard error

Answer 277

Dude no

Answer 278

?

Answer 279

I mean, what is the confidence interval.

Answer 280

\[29.71 \pm 1.96 (7.25624291383)\]

Answer 281

I'd use the full standard error

Answer 282

Yes?

Answer 283

See how there's two numbers here: [4.82, 5.18]

Do the math, mean plus z score of 1.96 times the standard error. Then do the same but minus. This will give you the two values for your brackets. Effectively these are the scores which would indicate a significant change statistically.

Answer 284

Um ... why? The question was whether we accept or reject the hypothesis and we figured that out. So, do I have to do what you jut said?

Answer 285

The gist of it is that you'd calculate a z score and if it's over z = +- 1.96 then it would fall in the last 5% on either end of the distribution, which means that you could reject the null hypothesis (h0 says that the hypothesis isn't true), so when you reject it, you're saying your hypothesis is correct. 5% is a number chosen in science because it's basically saying 'it's highly likely that we're solid with this hypothesis.'

The z score wasn't over 1.96 negative or positive so we're accepting H0 / rejecting H1 aka accepting the null hypothesis (that there wasn't a statistically significant change ).

Answer 286

Reading this after two years and I truly recognize how unprofessional my professor was. He would not answer questions and relied on the TA for everything. It was so late and I was losing my mind when I asked this question! Shadow is a real one for helping me with this.