A population of N =16 scores has a mean of p = 20, After one score is removed from the population, the new mean is found to be μ=19. What is the value of ine score that was removed? (Hint: Compare the val- ues for X before and after the score was removed.)

Question
A population of N =16 scores has a mean of p = 20, After one score is removed from the population, the new mean is found to be $$\displaystyleμ={19}$$. What is the value of ine score that was removed? (Hint: Compare the val- ues for X before and after the score was removed.)

2020-10-28
Let X1, X2,..., X16 be the starting population. We are remowing X16 from this population to obtain the second population.
Thus, the first mean is
PSK((Σ^16)/16)Xi)=20->Σ16Xi=320 i=1 i=1ZSK
The second mean is
PSK((Σ^15)/15)Xi)=19->Σ16Xi=285 i=1 i=1ZSK
From (1) we get that
PSKΣ^15X1+X16=320 i=1ZSK
Plug (2) into the above equation:
$$\displaystyle{285}+{X}{16}={320}\to{X}{16}={35}$$

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Previous studies show that $$\sigma_1 = 19$$.
For Englewood (a suburb of Denver), a random sample of $$n_2 = 12$$ winter days gave a sample mean pollution index of $$x_2 = 37$$.
Previous studies show that $$\sigma_2 = 13$$.
Assume the pollution index is normally distributed in both Englewood and Denver.
(a) State the null and alternate hypotheses.
$$H_0:\mu_1=\mu_2.\mu_1>\mu_2$$
$$H_0:\mu_1<\mu_2.\mu_1=\mu_2$$
$$H_0:\mu_1=\mu_2.\mu_1<\mu_2$$
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The standard normal. We assume that both population distributions are approximately normal with unknown standard deviations.
The standard normal. We assume that both population distributions are approximately normal with known standard deviations.
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At the $$\alpha = 0.01$$ level, we fail to reject the null hypothesis and conclude the data are not statistically significant.
At the $$\alpha = 0.01$$ level, we reject the null hypothesis and conclude the data are statistically significant.
At the $$\alpha = 0.01$$ level, we fail to reject the null hypothesis and conclude the data are statistically significant.
At the $$\alpha = 0.01$$ level, we reject the null hypothesis and conclude the data are not statistically significant.
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Reject the null hypothesis, there is sufficient evidence that there is a difference in mean pollution index for Englewood and Denver.
Fail to reject the null hypothesis, there is insufficient evidence that there is a difference in mean pollution index for Englewood and Denver.
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$$\mu_1 - \mu_2$$.
lower limit
upper limit
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Because the interval contains both positive and negative numbers, this indicates that at the 99% confidence level, the mean population pollution index for Englewood is greater than that of Denver.
Because the interval contains only negative numbers, this indicates that at the 99% confidence level, the mean population pollution index for Englewood is less than that of Denver.
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State the null and alternate hypotheses.
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What are the degrees of freedom?
$$df_{N} = ?$$
$$df_{D} = ?$$
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The populations follow independent normal distributions. We have random samples from each population.The populations follow dependent normal distributions. We have random samples from each population.The populations follow independent normal distributions.The populations follow independent chi-square distributions. We have random samples from each population.
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At the ? = 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.At the ? = 0.05 level, we fail to reject the null hypothesis and conclude the data are statistically significant. At the ? = 0.05 level, we reject the null hypothesis and conclude the data are not statistically significant.At the ? = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant.
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