The image presents a problem related to the distribution of fish in a lake, investigated by the Fish and Game Department. Initially, the fish were stocked with the following proportions: 30% catfish, 15% bass, 40% bluegill, and 15% pike. Five years later, a sample showed a distribution of:- Catfish: 124- Bass: 91- Bluegill: 208- Pike: 77The exercise involves testing whether the distribution of fish has changed at a significance level of 0.05. Here's a transcription of the tasks outlined:### (a) Determining the level of significance1. State the null and alternate hypotheses: - **H₀:** The distributions are the same. - **H₁:** The distributions are different.2. Multiple choice options for different hypotheses are listed, from which the correct pairing should be selected.### (b) Calculating the chi-square statistic1. Compute the chi-square statistic for the sample. Ensure expected frequencies are rounded to at least three decimal places.2. Check if all expected frequencies are greater than 5: - Yes/No option.3. Select the appropriate sampling distribution: - Options include normal, uniform, binomial, chi-square, Student’s t.4. Determine the degrees of freedom.### (c) P-value estimation- Find or estimate the P-value of the sample test statistic and round the answer to three decimal places.### (d) Decision regarding the null hypothesis- Based on previous answers, determine whether to reject or fail to reject the null hypothesis: - Depending on whether P-value > α or P-value ≤ α.### (e) Interpretation of results- Conclude whether there is sufficient evidence at the 5% level of significance to state if the fish distribution has changed from five years ago.

a) The proportion of Catfish, Bass, Bluegill and Pike in the Lake are 30%, 15%, 40%, and 15%,…

The Fish and Game Department stocked a lake with fish in the following proportions: 30% catfish, 15% bass, 40% bluegill, and 15% pike. Five years later it sampled the lake to see if the distribution of fish had changed. It found that the 500 fish in the sample were distributed as follows. Catfish 124 Bluegill 208 Pike 77 Bass 91 In the 5-year interval, did the distribution of fish change at the 0.05 level? (a) What is the level of significance? State the null and alternate hypotheses. O Mo: The distributions are different. H: The distributions are the same. O Ho: The distributions are the same. H: The distributions are the same. O Ho: The distributions are different. H: The distributions are different. O Ho: The distributions are the same. H: 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? O Yes O No What sampling distribution will you use? O normal O uniform O binomial O chi-square O Student's t What are the degrees of freedom? (c) Find or estimate the P-value of the sample test statistic. (Round your answer to three decimal places.) (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? O Since the P.value > a, we fail to reject the null hypothesis. O since the Pvalue > a, we reject the null hypothesis. O since the P-value s a, we reject the null hypothesis. O Since the P.value s a, we fail to reject the null hypothesis. (e) Interpret your conclusion in the context of the application. O At the 5% level of significance, the evidence is insufficient to conclude that current fish distribution is different than that of five years ago. O At the 5% level of significance, the evidence is sufficient to conclude that current fish distribution is different than that of five years ago.