Q25.  A dice is tossed 120 times with following results.     No. turned up: 1  2  3  4  5  6                    Frequency:        30 25 18 10 22 15      Test the hypothesis that the dice is unbiased (χ^2=11.7) . Calculate the frequency  observed for chi-square distribution*Dice is unbiased ,11.3Dice is biased ,12.9Dice is unbiased ,10.9Dice is biased ,12.3

Q22.   Suppose a person has 8 red , 5 green, 15 orange and 12 blue balls.Test the null hypothesis that the  colors of the balls occure with equal frequency. What is Chi Square value you get?*5.65.685.865.8

To calculate the chi-squared statistic, we need to follow these steps: ### Step-by-Step Calculation: #### iii) Calculate the Expected FrequenciesThe expected frequency for each cell in a contingency table is calculated using the formula: \[ E_{ij} = \frac{( \text{Row Total}_i \times \text{Column Total}_j )}{\text{Grand Total}} \] Let's calculate the expected frequencies for each cell: 1. **Facebook:** - Female: \( E_{11} = \frac{(117 \times 64)}{249} \approx 30.07 \) - Male: \( E_{12} = \frac{(132 \times 64)}{249} \approx 33.93 \) 2. **Instagram:** - Female: \( E_{21} = \frac{(117 \times 64)}{249} \approx 30.07 \) - Male: \( E_{22} = \frac{(132 \times 64)}{249} \approx 33.93 \) 3. **Snapchat:** - Female: \( E_{31} = \frac{(117 \times 58)}{249} \approx 27.25 \) - Male: \( E_{32} = \frac{(132 \times 58)}{249} \approx 30.75 \) 4. **Twitter:** - Female: \( E_{41} = \frac{(117 \times 63)}{249} \approx 29.6 \) - Male: \( E_{42} = \frac{(132 \times 63)}{249} \approx 33.4 \) The expected frequencies are: | | Facebook | Instagram | Snapchat | Twitter | Row Total | |------------|----------|-----------|----------|---------|-----------| | **Female** | 30.07 | 30.07 | 27.25 | 29.6 | 117 | | **Male** | 33.93 | 33.93 | 30.75 | 33.4 | 132 | | **Column Total** | 64 | 64 | 58 | 63 | 249 | #### iv) Calculate the Chi-Squared StatisticThe chi-squared statistic is calculated using the formula: \[ \chi^2 = \sum \frac{(O_{ij} - E_{ij})^2}{E_{ij}} \] Where \( O_{ij} \) is the observed frequency and \( E_{ij} \) is the expected frequency. Let's calculate the chi-squared statistic step by step: 1. **Facebook:** - Female: \( \frac{(33 - 30.07)^2}{30.07} \approx 0.29 \) - Male: \( \frac{(31 - 33.93)^2}{33.93} \approx 0.25 \) 2. **Instagram:** - Female: \( \frac{(30 - 30.07)^2}{30.07} \approx 0.00 \) - Male: \( \frac{(34 - 33.93)^2}{33.93} \approx 0.00 \) 3. **Snapchat:** - Female: \( \frac{(26 - 27.25)^2}{27.25} \approx 0.06 \) - Male: \( \frac{(32 - 30.75)^2}{30.75} \approx 0.05 \) 4. **Twitter:** - Female: \( \frac{(28 - 29.6)^2}{29.6} \approx 0.09 \) - Male: \( \frac{(35 - 33.4)^2}{33.4} \approx 0.08 \) Summing these values: \[ \chi^2 = 0.29 + 0.25 + 0.00 + 0.00 + 0.06 + 0.05 + 0.09 + 0.08 = 0.82 \] So, the chi-squared statistic is: \[ \chi^2 \approx 0.82 \] This value should be entered in the box for the chi-squared statistic.

A necessary assumption that is made when conducting a chi-squared analysis is: (select the one false statement)Group of answer choicesat least 80% of the observed frequencies are greater than or equal to 5all expected frequencies are greater than or equal to 1the sample size is large enough to approximate a Binomial random variable by a normal random variablethe sample data consist of frequency counts for different categories of a tablea subject can only fit into one category of the table (i.e. they belong with only one cell of the table)

The table shows the probabilities that a biased dice will land on 1, on 2, on 3, on 4, on 5and on 6Number on dice 1 2 3 4 5 6Probability 0.06 0.15 0.32 0.1 0.24 0.13Sven rolls the biased dice 400 times.Work out an estimate for the number of times the dice will land on 2 or on 5......................................................(Total for Question 6 is 2 marks)

Suppose that χ2 follows a chi-square distribution with 28 degrees of freedom.Use the ALEKS calculator to answer the following.(a) Compute P≤χ216. Round your answer to at least three decimal places.=P≤χ216(b) Find k such that =P≥χ2k0.05. Round your answer to at least two decimal places.=k