Question

One​ day, a​ city's baseball team lost their 78th game of the season and tied the​ city's major sport record for most consecutive losing seasons at 16. A losing season occurs when a team loses more games in a season than it wins. A typical baseball season consists of 154 ​games, so for the team to end their​ streak, they need to win 77 games in a season. Complete parts a through c below.

Click the icon to view the standard normal table.

a. Over the course of the​ streak, the team won approximately 41% of their games. For​ simplicity, assume the number of games they win in a given season follows a binomial distribution with n equals=154 and p equals=0.41 What is their expected number of wins in a​ season?

b. What is the probability that the team will win at least 77 games in a given​ season? (You may use technology to find the exact binomial probability or use the normal distribution to approximate the probability by finding a​ z-score for 81 and then evaluating the appropriate area under the normal​ curve.)

c. Can you think of any factors that might make the binomial distribution an inappropriate model for the number of games won in a​ season? Select any that could apply below. (different stadiums, price of tickets, weather, uniform colors, & different players)

Answer 1

a.

Expected number of wins in a​ season = n * p

= 154 * 0.41 = 63.14

b.

Standard deviation of number of wins in a​ season =

= 6.103491

Using Normal approximation to Binomial distribution, X ~ N( = 63.14 , = 6.103491)

Probability that the team will win at least 77 games in a given​ season = P(X 77)

= P[Z (77 - 63.14) / 6.103491]

= P[Z 2.27]

= 0.0116

c.

For binomial distribution model, the probability to win a game is not constant and independent and depends on the factors such as different stadiums, weather, & different players.