The types of raw materials used to construct stone tools found at an archaeological site are shown below. A random sample of 1486 stone tools were obtained from a current excavation site. Raw Material Regional Percent of Stone Tools Observed Number of Tools as Current excavation Site Basalt 61.3% 908 Obsidian 10.6% 166 Welded Tuff 11.4% 165 Pedernal chert 13.1% 198 Other 3.6% 49 Use a 1% level of significance to test the claim that the regional distribution of raw materials fits the distribution at the current excavation site. (a) What is the level of significance? State the null and alternate hypotheses. H0: The distributions are different. H1: The distributions are the same.H0: The distributions are the same. H1: The distributions are the same. H0: The distributions are different. H1: The distributions are different.H0: The distributions are the same. H1: The distributions are different. (b) Find the value of the chi-square statistic for the sample. (Round the expected frequencies to at least three decimal places. Round the test statistic to three decimal places.) Are all the expected frequencies greater than 5? YesNo What sampling distribution will you use? normal chi-square uniform Student's t binomial What are the degrees of freedom? (c) Find or estimate the P-value of the sample test statistic. P-value > 0.1000. 050 < P-value < 0.100 0.025 < P-value < 0.0500 .010 < P-value < 0.0250. 005 < P-value < 0.010 P-value < 0.005 (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis of independence? Since the P-value > ?, we fail to reject the null hypothesis. Since the P-value > ?, we reject the null hypothesis. Since the P-value ≤ ?, we reject the null hypothesis. Since the P-value ≤ ?, we fail to reject the null hypothesis. (e) Interpret your conclusion in the context of the application. At the 0.01 level of significance, the evidence is sufficient to conclude that the regional distribution of raw materials does not fit the distribution at the current excavation site. At the 0.01 level of significance, the evidence is insufficient to conclude that the regional distribution of raw materials does not fit the distribution at the current excavation site.
Addition Rule of Probability
It simply refers to the likelihood of an event taking place whenever the occurrence of an event is uncertain. The probability of a single event can be calculated by dividing the number of successful trials of that event by the total number of trials.
Expected Value
When a large number of trials are performed for any random variable ‘X’, the predicted result is most likely the mean of all the outcomes for the random variable and it is known as expected value also known as expectation. The expected value, also known as the expectation, is denoted by: E(X).
Probability Distributions
Understanding probability is necessary to know the probability distributions. In statistics, probability is how the uncertainty of an event is measured. This event can be anything. The most common examples include tossing a coin, rolling a die, or choosing a card. Each of these events has multiple possibilities. Every such possibility is measured with the help of probability. To be more precise, the probability is used for calculating the occurrence of events that may or may not happen. Probability does not give sure results. Unless the probability of any event is 1, the different outcomes may or may not happen in real life, regardless of how less or how more their probability is.
Basic Probability
The simple definition of probability it is a chance of the occurrence of an event. It is defined in numerical form and the probability value is between 0 to 1. The probability value 0 indicates that there is no chance of that event occurring and the probability value 1 indicates that the event will occur. Sum of the probability value must be 1. The probability value is never a negative number. If it happens, then recheck the calculation.
The types of raw materials used to construct stone tools found at an archaeological site are shown below. A random sample of 1486 stone tools were obtained from a current excavation site.
| Raw Material | Regional Percent of Stone Tools | Observed Number of Tools as Current excavation Site |
| Basalt | 61.3% | 908 |
| Obsidian | 10.6% | 166 |
| Welded Tuff | 11.4% | 165 |
| Pedernal chert | 13.1% | 198 |
| Other | 3.6% | 49 |
Use a 1% level of significance to test the claim that the regional distribution of raw materials fits the distribution at the current excavation site.
State the null and alternate hypotheses.
H1: The distributions are the same.H0: The distributions are the same.
H1: The distributions are the same. H0: The distributions are different.
H1: The distributions are different.H0: The distributions are the same.
H1: The distributions are different.
(b) Find the value of the chi-square statistic for the sample. (Round the expected frequencies to at least three decimal places. Round the test statistic to three decimal places.)
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 of independence?
(e) Interpret your conclusion in the context of the application.
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