Question

Please provide the following information. (a) What is the level of significance? State the null and alternate hypotheses. (b) Find the value of the chi-square statistic for the sample. Are all the expected frequencies greater than $5 ?$ What sampling distribution will you use? What are the degrees of freedom? (c) Find or estimate the $P$ -value of the sample test statistic. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis that the population fits the specified distribution of categories? (e) Interpret your conclusion in the context of the application. Meteorology: Normal Distribution The following problem is based on information from the National Oceanic and Atmospheric Administration (NOAA) Environmental Data Service. Let $x$ be a random variable that represents the average daily temperature (in degrees Fahrenheit) in July in the town of Kit Carson, Colorado. The $x$ distribution has a mean $\mu$ of approximately $75^{\circ} \mathrm{F}$ and standard deviation $\sigma$ of approximately $8^{\circ} \mathrm{F}$. A 20 -year study $(620$ July days) gave the entries in the rightmost column of the following table. (i) Remember that $\mu=75$ and $\sigma=8 .$ Examine Figure $6-5$ in Chapter $6 .$ Write a bricf explanation for Columns I, II, and III in the context of this problem. (ii) Use a $1 \%$ level of significance to test the claim that the average daily July temperature follows a normal distribution with $\mu=75$ and $\sigma=8$

Name: Problem 9, Chapter 10: Understandable Statistics, Concepts and Methods
Uploaded: 2021-02-23T06:04:26-08:00
Duration: 17 min 37 s
Channel: Harsh Gadhiya

   Please provide the following information.
(a) What is the level of significance? State the null and alternate hypotheses.
(b) Find the value of the chi-square statistic for the sample. Are all the expected frequencies greater than $5 ?$ What sampling distribution will you use? What are the degrees of freedom?
(c) Find or estimate the $P$ -value of the sample test statistic.
(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis that the population fits the specified distribution of categories?
(e) Interpret your conclusion in the context of the application.
Meteorology: Normal Distribution The following problem is based on information from the National Oceanic and Atmospheric Administration (NOAA) Environmental Data Service. Let $x$ be a random variable that represents the average daily temperature (in degrees Fahrenheit) in July in the town of Kit Carson, Colorado. The $x$ distribution has a mean $\mu$ of approximately $75^{\circ} \mathrm{F}$ and standard deviation $\sigma$ of approximately $8^{\circ} \mathrm{F}$. A 20 -year study $(620$ July days) gave the entries in the rightmost column of the following table.
(i) Remember that $\mu=75$ and $\sigma=8 .$ Examine Figure $6-5$ in Chapter $6 .$ Write a bricf explanation for Columns I, II, and III in the context of this problem.
(ii) Use a $1 \%$ level of significance to test the claim that the average daily July temperature follows a normal distribution with $\mu=75$ and $\sigma=8$

Understandable Statistics, Concepts and Methods

Charles Henry Brase,… 12th Edition

Chapter 10, Problem 9

Instant Answer

Solved by Expert Harsh Gadhiya

02/23/2021

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The problem is about the average daily temperature in July in the town of Kit Carson, Colorado. The data is given in a table with three columns. The first column represents the range of temperatures in terms of standard deviations from the mean. The second column Show more…

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Please provide the following information. (a) What is the level of significance? State the null and alternate hypotheses. (b) Find the value of the chi-square statistic for the sample. Are all the expected frequencies greater than $5 ?$ What sampling distribution will you use? What are the degrees of freedom? (c) Find or estimate the $P$ -value of the sample test statistic. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis that the population fits the specified distribution of categories? (e) Interpret your conclusion in the context of the application. Meteorology: Normal Distribution The following problem is based on information from the National Oceanic and Atmospheric Administration (NOAA) Environmental Data Service. Let $x$ be a random variable that represents the average daily temperature (in degrees Fahrenheit) in July in the town of Kit Carson, Colorado. The $x$ distribution has a mean $\mu$ of approximately $75^{\circ} \mathrm{F}$ and standard deviation $\sigma$ of approximately $8^{\circ} \mathrm{F}$. A 20 -year study $(620$ July days) gave the entries in the rightmost column of the following table. (i) Remember that $\mu=75$ and $\sigma=8 .$ Examine Figure $6-5$ in Chapter $6 .$ Write a bricf explanation for Columns I, II, and III in the context of this problem. (ii) Use a $1 \%$ level of significance to test the claim that the average daily July temperature follows a normal distribution with $\mu=75$ and $\sigma=8$

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Key Concepts

P-Value

The p-value is the probability of obtaining a test statistic at least as extreme as the one computed from the sample data, assuming that the null hypothesis is true. It quantifies the evidence against the null hypothesis; a small p-value indicates strong evidence against H0, leading to its rejection.

Interpretation of Statistical Results

Interpreting statistical results involves translating the outcome of the test into context-specific conclusions. This means explaining in practical terms what rejecting or failing to reject the null hypothesis implies regarding the distribution of the population, taking into account the specific application and the consequences of the decision made.

Degrees of Freedom

Degrees of freedom in a statistical test represent the number of independent values or quantities which can vary in an analysis without violating any constraints. For a chi-square goodness-of-fit test, the degrees of freedom are typically calculated as the number of categories minus the number of parameters estimated from the data minus one.

Expected Frequencies

Expected frequencies are the theoretical counts for each category based on the hypothesized distribution. A key assumption in the chi-square test is that these expected frequencies should be sufficiently large (commonly greater than 5) to ensure the validity of the approximation by the chi-square distribution.

Decision Making (Reject or Fail to Reject H0)

Based on the significance level and the p-value, statisticians decide whether to reject or fail to reject the null hypothesis. If the p-value is less than the significance level, it suggests that the observed distribution is unlikely under H0, and the null hypothesis is rejected, supporting the alternative hypothesis. Conversely, if the p-value is greater than or equal to the significance level, there is insufficient evidence to reject H0.

Null and Alternative Hypotheses

The null hypothesis (H0) is a statement that assumes no effect or no difference, and it serves as the default claim to be tested. The alternative hypothesis (Ha) is a statement that contradicts the null hypothesis, suggesting the presence of an effect or a difference. In goodness-of-fit tests, H0 typically asserts that the observed frequencies follow a specified distribution, while Ha indicates they do not.

Hypothesis Testing

Hypothesis testing is a statistical procedure used to decide whether there is enough evidence in a sample to infer that a certain condition holds for the entire population. It involves the formulation of null and alternative hypotheses, selection of a significance level, calculation of an appropriate test statistic based on the sample data, and comparison of the test statistic to a critical value or the computation of a p-value to make an inference.

Chi-Square Goodness-of-Fit Test

The Chi-Square Goodness-of-Fit Test is used to determine whether there is a significant difference between the expected frequencies and the observed frequencies in one or more categories. This test helps to assess how well the observed data fit a particular theoretical distribution, such as the normal distribution in many applications.

Significance Level

The significance level, often denoted by ?, is the probability of rejecting the null hypothesis when it is actually true. It represents the threshold for determining whether the observed data are extreme under the null hypothesis. A lower significance level (e.g., 1%) implies stricter criteria for rejecting the null hypothesis.

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Transcript

00:01 Now, what do we have in this question? they are saying let x be a random variable that represents the average daily temperatures in the month of july in a small town in colorado.

00:13 Okay.

00:14 Now, the x distribution has a mean of 75 degrees fahrenheit.

00:19 So, the mean, the mean is 75 degrees fahrenheit.

00:29 Okay.

00:32 And the standard deviation, sigma is approximately 8 degrees fahrenheit.

00:40 Okay, now a study was conducted over a span of 20 years, that is 620 july days.

00:48 And we are given a table of those entries from that study.

00:55 Okay, so what are the columns that we have? let us just look at the columns.

00:59 This we are also going to use in order to find the kye square statistic okay so here it is given that mu minus three sigma is less than equal to x which is less than mu minus two sigma mu minus 2 sigma okay similarly here i think there must be nu minus 2 sigma view minus sigma then there is mu there is mu plus sigma and there must be mu plus two sigma okay these are sigmas this is sigma this is sigma this is sigma saw than equal to less than equal to less than equal to less than equal to okay, then these are the lower limits and now we will write the upper limits.

02:03 This is going to be sigma.

02:05 This is going to be mu plus sigma.

02:08 This is going to be mu plus two sigma.

02:11 And this is going to be mu plus three sigma.

02:13 Okay, so what is happening over here is we are having a range, the frequency, the frequency count for the number of days when the temperature, was between mu minus 3 sigma and mu minus 2 sigma, right? when we can say that, okay, if i see this normal distribution, okay, let me just draw this normal distribution here.

02:44 This is the mean, okay? this is one standard deviation away.

02:50 This is two standard deviations away.

02:54 And this, if i extend the graph, this is going to be three standard deviations.

03:01 Survey right so all of these categories are basically the frequencies of how many values lie let's say between the first category is mu minus 3 sigma and mu minus 2 sigma so mu minus 3 sigma is 1 2 3 that is this point to mu minus 2 sigma is this point so what is the frequency that lies between this okay the number of values that lie in this region then the number of values that lie between mu minus 2 sigma and mu minus sigma right that is this region so in this way this is the study of all the different regions.

03:34 All right.

03:35 So this is the first column.

03:39 The second column is actually the values.

03:43 What is sigma? sigma is 8.

03:46 So what is happening over here is now they have just given us the values.

03:53 When x is between 51 to 59.

04:00 Between 51 to 59.

04:02 Then when x is between 59 to 67.

04:10 Then when x is between 67 to 75 then when x is between 75 to 83 then when x is between 83 to 91 then when x is between 91 to 99 okay so these are the temperatures right what is the the range when x minus 3 sigma we do we get 51 degrees and when we do x minus 2 sigma we get 99 degrees so what was the frequency or what was the number of days when the temperature was between these two values right so this is just a study of the number of values that are between 2 and 3 standard deviations away to the left then 1 and 2 standard deviations away to the left and so on okay now we are i think we are also given the expected percent from normal curve okay now what is happening is they are expecting this to be perfectly normal, right? this distribution to be perfectly normal.

05:18 So this is the third column.

05:21 Okay.

05:22 So what should be the values? if it is indeed perfectly normal, it should be 2 .35%, 2 .35 % for the first category.

05:30 For the second one, it should be 13 .5%.

05:35 Okay.

05:36 Then it should be 34%.

05:39 Then again 34%.

05:41 Then it should be 30%.

05:42 13 .5 % again and then it should be 2 .35%.

05:47 Because the normal distribution is symmetric, we can see that the values here are symmetric, right? 2 .35, 13 .5, 34.

05:56 Then in the decreasing order, 34, 13 .5 and 2 .35.

06:00 Okay.

06:03 Now this is expected.

06:06 Okay.

06:08 But what are the observed values? what are the observed? observed values now the observed values are there were 16 days when the temperature was between two and three standard deviations away to the left then for the second category it was 78 days then it was 212 days then it was 221 days then it was 81 days then it was 12 days okay now this addition is given to us a 620 which means that this is our sample size n this is our sample size n.

06:49 Okay.

06:51 Now what is going to be the expected value? in order to find the kai square statistic, we also need to find the expected values, expected values...

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