2) If 4 samples from 4 populations are given, then what is the probability of no Type 1 error at 99% confidence interval?

3) Pepsi Co. Pvt Ltd is planning to produce a new energy drink. For this purpose, a pilot study was conducted using four different energy drinks formula. They were tested on 3 different sets of people and the reactions in energy level of the people were measured forming four groups. Using the data (measured energy level values) provided below, test whether there is difference in the four energy drinks. Perform this by finding F ratio. What is F ratio at 95% CI?. The data is as shown in Table 1.

Formula 1 Formula 2 Formula 3 Formula 4

84 26 13 38

65 42 32 25

70 52 45 37

4) Pepsi Co. Pvt Ltd is planning to produce a new energy drink. For this purpose, a pilot study was conducted using four different energy drinks formula. They were tested on 3 different sets of people and the reactions in energy level of the people were measured forming four groups. Using the data (measured energy level values) provided below, test whether there is difference in the four energy drinks. Perform this by finding F ratio. What is F critical , i.e. Fc at 95% CI?. The data is as shown in Table 1.

Formula 1 Formula 2 Formula 3 Formula 4

84 26 13 38

65 42 32 25

70 52 45 37

5) Mr. Ali runs a gift shop in the Defense area, Karachi. He advertises weekly in the local newspapers and is considering increasing his advertising budget. Before doing so, he decides to evaluate the past effectiveness of these advertisements. Five weeks are sampled, and the advertising expenses (in PKR) and sales volume for each is shown in the Table 2. What is correlation coefficient value.

S.NO. Sales (PKR 100) Advertising(PKR 100)

1.00 4.00 4.10

2.00 7.00 6.72

3.00 6.00 5.05

4.00 8.00 7.26

5.00 4.00 2.69

6) Mr. Ali runs a gift shop in the Defense area, Karachi. He advertises weekly in the local newspapers and is considering increasing his advertising budget. Before doing so, he decides to evaluate the past effectiveness of these advertisements. Five weeks are sampled, and the advertising expenses (in PKR) and sales volume for each is shown in the Table 2. Develop a regression equation that would help Mr. Ali evaluate his advertising. Find slope value.

S.NO. Sales (PKR 100) Advertising(PKR 100)

1.00 4.00 4.10

2.00 7.00 6.72

3.00 6.00 5.05

4.00 8.00 7.26

5.00 4.00 2.69

7) Mr. Ali runs a gift shop in the Defense area, Karachi. He advertises weekly in the local newspapers and is considering increasing his advertising budget. Before doing so, he decides to evaluate the past effectiveness of these advertisements. Five weeks are sampled, and the advertising expenses (in PKR) and sales volume for each is shown in the Table 2. Develop a regression equation that would help Mr. Ali evaluate his advertising. Find constant value.

S.NO. Sales (PKR 100) Advertising(PKR 100)

1.00 4.00 4.10

2.00 7.00 6.72

3.00 6.00 5.05

4.00 8.00 7.26

5.00 4.00 2.69

Mr. Adam is a coffee addict. He keeps switching between three brands of coffee, say A, B and C, from week to week according to a discrete Markov chain with the following transition probability matrix: 0.2 0.3 0.57 P = 0.1 0 0.9 0.55 0 0.45 1. Check whether the Markov chain is an irreducible chain or not? 2. Is the Markov chain is periodic or aperiodic? 3. Is the Markov chain has a unique stationary distribution? 4. If Mr. Adam is using brand A this week (ie., week 1), what is the probability distribution of the brand he will be using in week 20?al d) Of the 1682 respondents, 493 identified themselves as Catholic. Statistic e) Four hundred thirty of the respondents reported that they had never married. Statistic (Noticing that a part of the calculation here corresponds to the calculation already done in d can save you a bit of work) f) The proportion of respondents who voted for the Conservative Party of Canada in the prior fed election was 0.36. Statistic (Noticing that a part of the calculation here corresponds to the calculation already dane in d can save you a but of work.)6. A 3-state Markov chain has the following transition matrix: 0.6 0.3 0.17 P = 0.2 0.7 0.1 0.5 0 0.5 Using Xo = (0.6, 0.0, 0.4), find X, and X2. 0.4 0.6 0 7. A 3-state Markov chain has transition matrix P = 0 0.6 0.4 0.6 0.2 0.2 Find the steady state vector for this Markov chain