Use a table to find or estimate χ 2 \chi^2 such that: P ( X 2 < χ 2 ) = 0.010 P\left(X^2<\chi^2\right)=0.010 (Area to the left ) d f = 50 df=50

All right, everyone. So let's take a look at this practice problem. Now, it looks very similar to the one that we just did. We're going to be using a table to find or estimate a chi squared value such that and we're given a statement that looks very similar to the one we just did before. We have p that x squared is less than some chi squared value, and that's equal to 0.01. Now, if you remember, this means that it's an area to the left and it changes everything because remember that when you use your chi squared tables, it's always assuming that the numbers that you're looking at over here are areas to the right of the chi squared value. So there's one really important step that we have to do first, which is we have to convert this area from a left area to a right one. And all you do is you're just basically going to be using the complement rule. So I just wanna be really careful. Don't use this value when you're looking at your chi squared. You're gonna have to convert this to an area to the right. And this is just gonna be one minus 0.01, which is gonna give you 0.99. Alright? So this is the area to the right of that chi squared value that we're interested in, and this is what we have to use. Alright? So just be very careful there. So let's actually go ahead and use our tables or you can go ahead and just do it on your own, but let's go ahead and do that. So in our chi squared value, remember that your higher areas like point nines are gonna be on the left side of that table, and we're gonna be looking at point nine nine over here. So it's going to go all the way down until we hit degrees of freedom of 50. Let's go ahead and do that. If you go all the way down here, all the way down until degrees of freedom of 50, you should hit this number over here, 29.707. Right. So that's our number. You go back to our original problem, and and we just write that this is 29.7. And, again, if you want to round this to the nearest hundredth place, that's fine. It's 29.71. So, again, just as a reminder, whenever you're given a left tilt area, whether you're finding a critical value or just trying to find a chi squared value, you always have to plug into that table or you always have to look up the value as if it was an area to the right. Alright. So just be careful there. Alright. Thanks for watching.

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8. Sampling Distributions & Confidence Intervals: Proportion

Chi Square Distribution

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8. Sampling Distributions & Confidence Intervals: Proportion

Chi Square Distribution

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Multiple Choice

Use a table to find or estimate $χ^{2}$ such that:
$P (X^{2} < χ^{2}) = 0.010$ (Area to the left)
$d f = 50$

27.99

29.71

76.15

79.49

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Verified step by step guidance

Identify the problem: You need to find the chi-square value $ \chi^2 $ such that the cumulative probability to the left of this value is 0.010, i.e., $ P(X^2 < \chi^2) = 0.010 $, with degrees of freedom $ df = 50 $.

Understand that chi-square tables typically provide the right-tail probabilities $ P(X^2 > \chi^2) $. Since you want the left-tail probability, convert it to the right-tail probability using the complement rule: $ P(X^2 > \chi^2) = 1 - 0.010 = 0.990 $.

Locate the row in the chi-square table corresponding to $ df = 50 $. Then, find the chi-square value that corresponds to a right-tail probability of 0.990 (which matches the left-tail probability of 0.010).

Read off or interpolate the chi-square value from the table for $ df = 50 $ and right-tail probability 0.990. This value is the $ \chi^2 $ such that $ P(X^2 < \chi^2) = 0.010 $.

Verify your answer by checking that the cumulative probability to the left of the found $ \chi^2 $ value is indeed 0.010, either by using the table or a chi-square distribution calculator.